mirror of https://gitlab.com/rnger/amath
1858 lines
35 KiB
Groff
1858 lines
35 KiB
Groff
.TH "amath/amathi.h" 3 "Wed Mar 1 2017" "Version 1.7.1" "amath" \" -*- nroff -*-
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.ad l
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.nh
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.SH NAME
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amathi.h \- Complex numbers math library
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.SH SYNOPSIS
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.br
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.PP
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.SS "Classes"
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.in +1c
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.ti -1c
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.RI "union \fBcomplex\fP"
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.br
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.in -1c
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.SS "Functions"
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.in +1c
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.ti -1c
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.RI "double \fBcsgn\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIComplex signum\&. \fP"
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.ti -1c
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.RI "double \fBcabs\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIAbsolute value of complex number\&. \fP"
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.ti -1c
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.RI "double \fBcreal\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIReal part of complex number\&. \fP"
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.ti -1c
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.RI "double \fBcimag\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIImaginary part of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcexp\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIReturns e to the power of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBclog\fP (\fBcomplex\fP z)"
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.br
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.RI "\fINatural logarithm of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBconj\fP (\fBcomplex\fP z)"
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.br
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.ti -1c
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.RI "\fBcomplex\fP \fBcsqrt\fP (\fBcomplex\fP z)"
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.br
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.RI "\fISquare root of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcpow\fP (\fBcomplex\fP x, \fBcomplex\fP z)"
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.br
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.RI "\fIComplex number raised to a power\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcpack\fP (double x, double y)"
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.br
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.RI "\fIPack two real numbers into a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcadd\fP (\fBcomplex\fP a, \fBcomplex\fP z)"
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.br
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.RI "\fIAddition of two complex numbers\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcsub\fP (\fBcomplex\fP a, \fBcomplex\fP z)"
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.br
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.RI "\fISubtraction of two complex numbers\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcmul\fP (\fBcomplex\fP a, \fBcomplex\fP z)"
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.br
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.RI "\fIMultiplication of two complex numbers\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcdiv\fP (\fBcomplex\fP a, \fBcomplex\fP z)"
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.br
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.RI "\fIDivision of two complex numbers\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcceil\fP (\fBcomplex\fP z)"
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.br
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.RI "\fICeiling value of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcfloor\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIFloor value of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBctrunc\fP (\fBcomplex\fP z)"
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.br
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.RI "\fITruncated value of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcround\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIDivision of two complex numbers\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcreci\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIReciprocal value of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccbrt\fP (\fBcomplex\fP z)"
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.br
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.RI "\fICube root of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBclogb\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIBase 2 logarithmic value of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBclog10\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIBase 10 logarithmic value of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccos\fP (\fBcomplex\fP z)"
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.br
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.RI "\fICosine of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcsin\fP (\fBcomplex\fP z)"
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.br
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.RI "\fISine of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBctan\fP (\fBcomplex\fP z)"
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.br
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.RI "\fITangent of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcsec\fP (\fBcomplex\fP z)"
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.br
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.RI "\fISecant of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccsc\fP (\fBcomplex\fP z)"
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.br
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.RI "\fICosecant of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccot\fP (\fBcomplex\fP z)"
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.br
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.RI "\fICotangent of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcacos\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse cosine of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcasin\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse sine of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcatan\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse tangent of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcasec\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse secant expressed using complex logarithms: \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcacsc\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse cosecant of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcacot\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse cotangent of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccosh\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIHyperbolic cosine of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcsinh\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIHyperbolic sine of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBctanh\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIHyperbolic tangent of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcsech\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIHyperbolic secant of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccsch\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIHyperbolic secant of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBccoth\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIHyperbolic cotangent of a complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcacosh\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse hyperbolic cosine of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcasinh\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse hyperbolic sine of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcatanh\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse hyperbolic tangent of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcasech\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse hyperbolic secant of complex numbers\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcacsch\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse hyperbolic cosecant of complex number\&. \fP"
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.ti -1c
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.RI "\fBcomplex\fP \fBcacoth\fP (\fBcomplex\fP z)"
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.br
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.RI "\fIInverse hyperbolic cotangent of complex number\&. \fP"
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.in -1c
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.SH "Detailed Description"
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.PP
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Functions in math library for handling complex numbers\&.
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.PP
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Definition in file \fBcomplex\&.h\fP\&.
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.SH "Function Documentation"
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.PP
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.SS "double cabs (\fBcomplex\fP z)"
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.PP
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Absolute value of complex number\&.
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.PP
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Definition at line 59 of file prim\&.c\&.
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.PP
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References cimag(), creal(), and hypot()\&.
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.PP
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.nf
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60 {
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61 return hypot(creal(z), cimag(z));
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62 }
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.fi
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.SS "\fBcomplex\fP cacos (\fBcomplex\fP z)"
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.PP
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Inverse cosine of complex number\&.
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.PP
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\fBVersion:\fP
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.RS 4
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1\&.0
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.RE
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.PP
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\fBDate:\fP
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.RS 4
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14/09/15
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.RE
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.PP
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Inverse cosine expressed using complex logarithms:
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.PP
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.nf
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arccos z = -i * log(z + i * sqrt(1 - z * z))
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.fi
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.PP
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More info is available at Wikipedia:
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.br
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https://wikipedia.org/wiki/Inverse_trigonometric_functions#Logarithmic_forms
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.PP
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Definition at line 46 of file cacos\&.c\&.
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.PP
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References cadd(), clog(), cmul(), cpack(), csqrt(), and csub()\&.
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.PP
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.nf
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47 {
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48 complex a = cpack(1\&.0, 0\&.0);
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49 complex i = cpack(0\&.0, 1\&.0);
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50 complex j = cpack(0\&.0, -1\&.0);
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51 complex p = csub(a, cmul(z, z));
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52 complex q = clog(cadd(z, cmul(i, csqrt(p))));
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53 complex w = cmul(j, q);
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54 return w;
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55 }
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.fi
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.SS "\fBcomplex\fP cacosh (\fBcomplex\fP z)"
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.PP
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Inverse hyperbolic cosine of complex number\&.
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.PP
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\fBVersion:\fP
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.RS 4
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1\&.1
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.RE
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.PP
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\fBDate:\fP
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.RS 4
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15/03/03
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.RE
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.PP
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Inverse hyperbolic cosine expressed using complex logarithms:
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.PP
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.nf
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acosh(z) = log(z + sqrt(z*z - 1))
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.fi
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.PP
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More info is available at Wikipedia:
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.br
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https://wikipedia.org/wiki/Inverse_hyperbolic_function#Logarithmic_representation
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.PP
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Definition at line 46 of file cacosh\&.c\&.
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.PP
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References cadd(), clog(), cmul(), cpack(), csqrt(), and csub()\&.
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.PP
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.nf
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47 {
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48 complex one = cpack(1\&.0, 0\&.0);
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49 complex a = csub(cmul(z, z), one);
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50 complex b = cadd(z, csqrt(a));
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51 complex w = clog(b);
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52 return w;
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53 }
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.fi
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.SS "\fBcomplex\fP cacot (\fBcomplex\fP z)"
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.PP
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Inverse cotangent of complex number\&.
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.PP
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\fBVersion:\fP
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.RS 4
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1\&.1
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.RE
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.PP
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\fBDate:\fP
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.RS 4
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14/10/01
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.RE
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.PP
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Inverse cotangent expressed using complex logarithms:
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.PP
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.nf
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arccot z = i/2 * (log(1 - i/z) - log(1 + i/z))
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.fi
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.PP
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More info is available at Wikipedia:
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.br
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https://wikipedia.org/wiki/Inverse_trigonometric_functions#Logarithmic_forms
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.PP
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Definition at line 46 of file cacot\&.c\&.
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.PP
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References cadd(), cdiv(), clog(), cmul(), cpack(), and csub()\&.
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.PP
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.nf
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47 {
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48 complex one = cpack(1\&.0, 0\&.0);
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49 complex two = cpack(2\&.0, 0\&.0);
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50 complex i = cpack(0\&.0, 1\&.0);
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51 complex iz = cdiv(i, z);
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52 complex p = clog(csub(one, iz));
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53 complex q = clog(cadd(one, iz));
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54 complex w = cmul(cdiv(i, two), csub(p, q));
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55 return w;
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56 }
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.fi
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.SS "\fBcomplex\fP cacoth (\fBcomplex\fP z)"
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.PP
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Inverse hyperbolic cotangent of complex number\&.
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.PP
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\fBVersion:\fP
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.RS 4
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1\&.0
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.RE
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.PP
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\fBDate:\fP
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.RS 4
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14/09/15
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.RE
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.PP
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Inverse hyperbolic cotangent expressed using complex logarithms:
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.PP
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.nf
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acoth(z) = 1/2 * ((log(z + 1) - log(z - 1))
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.fi
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.PP
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More info is available at Wikipedia:
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.br
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https://wikipedia.org/wiki/Inverse_hyperbolic_function#Logarithmic_representation
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.PP
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Definition at line 46 of file cacoth\&.c\&.
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.PP
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References cadd(), clog(), cmul(), cpack(), and csub()\&.
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.PP
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.nf
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47 {
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48 complex half = cpack(0\&.5, 0\&.0);
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49 complex one = cpack(1\&.0, 0\&.0);
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50 complex a = clog(cadd(z, one));
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51 complex b = clog(csub(z, one));
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52 complex c = csub(a, b);
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53 complex w = cmul(half, c);
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54 return w;
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55 }
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.fi
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.SS "\fBcomplex\fP cacsc (\fBcomplex\fP z)"
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.PP
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Inverse cosecant of complex number\&.
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.PP
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\fBVersion:\fP
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.RS 4
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1\&.1
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.RE
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.PP
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\fBDate:\fP
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.RS 4
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14/10/01
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.RE
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.PP
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Inverse cosecant expressed using complex logarithms:
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.PP
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.nf
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arccsc z = -i * log(sqr(1 - 1/(z*z)) + i/z)
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.fi
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.PP
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More info is available at Wikipedia:
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.br
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https://wikipedia.org/wiki/Inverse_trigonometric_functions#Logarithmic_forms
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.PP
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Definition at line 46 of file cacsc\&.c\&.
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.PP
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References cadd(), cdiv(), clog(), cmul(), cpack(), csqrt(), and csub()\&.
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.PP
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.nf
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47 {
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48 complex one = cpack(1\&.0, 0\&.0);
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49 complex i = cpack(0\&.0, 1\&.0);
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50 complex j = cpack(0\&.0, -1\&.0);
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51 complex iz = cdiv(i, z);
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52 complex z2 = cmul(z, z);
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53 complex p = cdiv(one, z2);
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54 complex q = csqrt(csub(one, p));
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55 complex w = cmul(j, clog(cadd(q, iz)));
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56 return w;
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57 }
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.fi
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.SS "\fBcomplex\fP cacsch (\fBcomplex\fP z)"
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.PP
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Inverse hyperbolic cosecant of complex number\&.
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.PP
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\fBVersion:\fP
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.RS 4
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1\&.0
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.RE
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.PP
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\fBDate:\fP
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.RS 4
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14/09/15
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.RE
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.PP
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Inverse hyperbolic cosecant expressed using complex logarithms:
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.PP
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.nf
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acsch(z) = log(sqrt(1 + 1 / (z * z)) + 1/z)
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.fi
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.PP
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More info is available at Wikipedia:
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.br
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https://wikipedia.org/wiki/Inverse_hyperbolic_function#Logarithmic_representation
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.PP
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Definition at line 46 of file cacsch\&.c\&.
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.PP
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References cadd(), clog(), cmul(), cpack(), creci(), and csqrt()\&.
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.PP
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.nf
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47 {
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48 complex one = cpack(1\&.0, 0\&.0);
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49 complex a = creci(cmul(z, z));
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50 complex b = csqrt(cadd(one, a));
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51 complex c = cadd(b, creci(z));
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|
52 complex w = clog(c);
|
|
53 return w;
|
|
54 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cadd (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Addition of two complex numbers\&.
|
|
.PP
|
|
Definition at line 131 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
132 {
|
|
133 complex w;
|
|
134 w = cpack(creal(y) + creal(z), cimag(y) + cimag(z));
|
|
135 return w;
|
|
136 }
|
|
.fi
|
|
.SS "\fBcomplex\fP casec (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Inverse secant expressed using complex logarithms:
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
14/10/01
|
|
.RE
|
|
.PP
|
|
Inverse secant expressed using complex logarithms:
|
|
.PP
|
|
.nf
|
|
|
|
arcsec z = -i * log(i * sqr(1 - 1/(z*z)) + 1/z)
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Inverse_trigonometric_functions#Logarithmic_forms
|
|
.PP
|
|
Definition at line 46 of file casec\&.c\&.
|
|
.PP
|
|
References cadd(), cdiv(), clog(), cmul(), cpack(), creci(), csqrt(), and csub()\&.
|
|
.PP
|
|
.nf
|
|
47 {
|
|
48 complex one = cpack(1\&.0, 0\&.0);
|
|
49 complex i = cpack(0\&.0, 1\&.0);
|
|
50 complex j = cpack(0\&.0, -1\&.0);
|
|
51 complex rz = creci(z);
|
|
52 complex z2 = cmul(z, z);
|
|
53 complex p = cdiv(one, z2);
|
|
54 complex q = csqrt(csub(one, p));
|
|
55 complex w = cmul(j, clog(cadd(cmul(i, q), rz)));
|
|
56 return w;
|
|
57 }
|
|
.fi
|
|
.SS "\fBcomplex\fP casech (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Inverse hyperbolic secant of complex numbers\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
15/03/03
|
|
.RE
|
|
.PP
|
|
Inverse hyperbolic secant expressed using complex logarithms:
|
|
.PP
|
|
.nf
|
|
|
|
asech(z) = log(sqrt(1 / (z * z) - 1) + 1/z)
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Inverse_hyperbolic_function#Logarithmic_representation
|
|
.PP
|
|
Definition at line 47 of file casech\&.c\&.
|
|
.PP
|
|
References cadd(), clog(), cmul(), cpack(), creci(), csqrt(), and csub()\&.
|
|
.PP
|
|
.nf
|
|
48 {
|
|
49 complex one = cpack(1\&.0, 0\&.0);
|
|
50 complex a = creci(cmul(z, z));
|
|
51 complex b = csqrt(csub(a, one));
|
|
52 complex c = cadd(b, creci(z));
|
|
53 complex w = clog(c);
|
|
54 return w;
|
|
55 }
|
|
.fi
|
|
.SS "\fBcomplex\fP casin (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Inverse sine of complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
14/10/01
|
|
.RE
|
|
.PP
|
|
Inverse sine expressed using complex logarithms:
|
|
.PP
|
|
.nf
|
|
|
|
arcsin z = -i * log(iz + sqrt(1 - z*z))
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Inverse_trigonometric_functions#Logarithmic_forms
|
|
.PP
|
|
Definition at line 46 of file casin\&.c\&.
|
|
.PP
|
|
References cadd(), clog(), cmul(), cpack(), csqrt(), and csub()\&.
|
|
.PP
|
|
.nf
|
|
47 {
|
|
48 complex one = cpack(1\&.0, 0\&.0);
|
|
49 complex i = cpack(0\&.0, 1\&.0);
|
|
50 complex j = cpack(0\&.0, -1\&.0);
|
|
51 complex iz = cmul(i, z);
|
|
52 complex z2 = cmul(z, z);
|
|
53 complex p = csqrt(csub(one, z2));
|
|
54 complex q = clog(cadd(iz, p));
|
|
55 complex w = cmul(j, q);
|
|
56 return w;
|
|
57 }
|
|
.fi
|
|
.SS "\fBcomplex\fP casinh (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Inverse hyperbolic sine of complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.0
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
14/09/15
|
|
.RE
|
|
.PP
|
|
Inverse hyperbolic sine expressed using complex logarithms:
|
|
.PP
|
|
.nf
|
|
|
|
asinh(z) = log(z + sqrt(z*z + 1))
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
With branch cuts: -i INF to -i and i to i INF
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
Domain: -INF to INF
|
|
Range: -INF to INF
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Inverse_hyperbolic_function#Logarithmic_representation
|
|
.PP
|
|
Definition at line 51 of file casinh\&.c\&.
|
|
.PP
|
|
References cadd(), clog(), cmul(), cpack(), and csqrt()\&.
|
|
.PP
|
|
.nf
|
|
52 {
|
|
53 complex one = cpack(1\&.0, 0\&.0);
|
|
54 complex a = cadd(cmul(z, z), one);
|
|
55 complex b = cadd(z, csqrt(a));
|
|
56 complex w = clog(b);
|
|
57 return w;
|
|
58 }
|
|
.fi
|
|
.SS "\fBcomplex\fP catan (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Inverse tangent of complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
14/10/01
|
|
.RE
|
|
.PP
|
|
Inverse tangent expressed using complex logarithms:
|
|
.PP
|
|
.nf
|
|
|
|
atan(z) = i/2 * (log(1 - i * z) - log(1 + i * z))
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Inverse_trigonometric_functions#Logarithmic_forms
|
|
.PP
|
|
Definition at line 46 of file catan\&.c\&.
|
|
.PP
|
|
References cadd(), cdiv(), clog(), cmul(), cpack(), and csub()\&.
|
|
.PP
|
|
.nf
|
|
47 {
|
|
48 complex one = cpack(1\&.0, 0\&.0);
|
|
49 complex two = cpack(2\&.0, 0\&.0);
|
|
50 complex i = cpack(0\&.0, 1\&.0);
|
|
51 complex iz = cmul(i, z);
|
|
52 complex p = clog(csub(one, iz));
|
|
53 complex q = clog(cadd(one, iz));
|
|
54 complex w = cmul(cdiv(i, two), csub(p, q));
|
|
55 return w;
|
|
56 }
|
|
.fi
|
|
.SS "\fBcomplex\fP catanh (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Inverse hyperbolic tangent of complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.0
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
14/09/15
|
|
.RE
|
|
.PP
|
|
Inverse hyperbolic tangent expressed using complex logarithms:
|
|
.PP
|
|
.nf
|
|
|
|
atanh(z) = 1/2 * ((log(1 + z) - log(1 - z))
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Inverse_hyperbolic_function#Logarithmic_representation
|
|
.PP
|
|
Definition at line 46 of file catanh\&.c\&.
|
|
.PP
|
|
References cadd(), clog(), cmul(), cpack(), and csub()\&.
|
|
.PP
|
|
.nf
|
|
47 {
|
|
48 complex half = cpack(0\&.5, 0\&.0);
|
|
49 complex one = cpack(1\&.0, 0\&.0);
|
|
50 complex a = clog(cadd(one, z));
|
|
51 complex b = clog(csub(one, z));
|
|
52 complex c = csub(a, b);
|
|
53 complex w = cmul(half, c);
|
|
54 return w;
|
|
55 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccbrt (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Cube root of complex number\&.
|
|
.PP
|
|
.nf
|
|
|
|
cbrt z = exp(1/3 * log(z))
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Cube_root
|
|
.PP
|
|
Definition at line 43 of file ccbrt\&.c\&.
|
|
.PP
|
|
References cexp(), clog(), cmul(), and cpack()\&.
|
|
.PP
|
|
.nf
|
|
44 {
|
|
45 complex onethird = cpack(1\&.0 / 3\&.0, 0\&.0);
|
|
46 complex a = cmul(onethird, clog(z));
|
|
47 complex w = cexp(a);
|
|
48 return w;
|
|
49 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cceil (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Ceiling value of complex number\&.
|
|
.PP
|
|
Definition at line 109 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
110 {
|
|
111 complex w;
|
|
112 w = cpack(ceil(creal(z)), ceil(cimag(z)));
|
|
113 return w;
|
|
114 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccos (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Cosine of complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
real = cos(a) * cosh(b)
|
|
imag = -sin(a) * sinh(b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 49 of file ccos\&.c\&.
|
|
.PP
|
|
References cchsh(), cimag(), cos(), cpack(), creal(), and sin()\&.
|
|
.PP
|
|
.nf
|
|
50 {
|
|
51 complex w;
|
|
52 double a, b;
|
|
53 double ch, sh;
|
|
54
|
|
55 a = creal(z);
|
|
56 b = cimag(z);
|
|
57 cchsh(b, &ch, &sh);
|
|
58 w = cpack((cos(a) * ch), (-sin(a) * sh));
|
|
59
|
|
60 return w;
|
|
61 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccosh (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Hyperbolic cosine of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
real = cosh(a) * cos(b)
|
|
imag = sinh(a) * sin(b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 52 of file ccosh\&.c\&.
|
|
.PP
|
|
References cchsh(), cimag(), cos(), cpack(), creal(), and sin()\&.
|
|
.PP
|
|
.nf
|
|
53 {
|
|
54 complex w;
|
|
55 double a, b;
|
|
56 double ch, sh;
|
|
57
|
|
58 a = creal(z);
|
|
59 b = cimag(z);
|
|
60 cchsh(a, &ch, &sh);
|
|
61 w = cpack(cos(b) * ch, sin(b) * sh);
|
|
62
|
|
63 return w;
|
|
64 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccot (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Cotangent of a complex number\&. Calculated as in Open Office:
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
sin(2\&.0 * a)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * b) - cos(2\&.0 * a)
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
-sinh(2\&.0 * b)
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * b) - cos(2\&.0 * a)
|
|
.fi
|
|
.PP
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMCOT_function
|
|
.PP
|
|
Definition at line 50 of file ccot\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 complex w;
|
|
53 double a, b;
|
|
54 double d;
|
|
55
|
|
56 a = creal(z);
|
|
57 b = cimag(z);
|
|
58 d = cosh(2\&.0 * b) - cos(2\&.0 * a);
|
|
59
|
|
60 if (d == 0\&.0)
|
|
61 {
|
|
62 w = cpack(D_INFP, D_INFP);
|
|
63 }
|
|
64 else
|
|
65 {
|
|
66 w = cpack((sin(2\&.0 * a) / d), (-sinh(2\&.0 * b) / d));
|
|
67 }
|
|
68
|
|
69 return w;
|
|
70 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccoth (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Hyperbolic cotangent of a complex number\&.
|
|
.PP
|
|
.nf
|
|
|
|
acoth(z) = 0\&.5 * (log(1 + 1/z) - log(1 - 1/z))
|
|
.fi
|
|
.PP
|
|
or
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
sinh(2\&.0 * a)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * a) - cos(2\&.0 * b)
|
|
.PP
|
|
.nf
|
|
-sin(2.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * a) - cos(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 52 of file ccoth\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
53 {
|
|
54 complex w;
|
|
55 double a, b;
|
|
56 double d;
|
|
57
|
|
58 a = creal(z);
|
|
59 b = cimag(z);
|
|
60 d = cosh(2\&.0 * a) - cos(2\&.0 * b);
|
|
61 w = cpack(sinh(2\&.0 * a) / d, -sin(2\&.0 * b) / d);
|
|
62
|
|
63 return w;
|
|
64 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccsc (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Cosecant of a complex number\&. Calculated as in Open Office:
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
2\&.0 * sin(a) * cosh(b)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * b) - cos(2\&.0 * a)
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
-2\&.0 * cos(a) * sinh(b)
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * b) - cos(2\&.0 * a)
|
|
.fi
|
|
.PP
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMCSC_function
|
|
.PP
|
|
Definition at line 50 of file ccsc\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 complex w;
|
|
53 double a, b;
|
|
54 double d;
|
|
55
|
|
56 a = creal(z);
|
|
57 b = cimag(z);
|
|
58 d = cosh(2\&.0 * b) - cos(2\&.0 * a);
|
|
59
|
|
60 if (d == 0\&.0)
|
|
61 {
|
|
62 w = cpack(D_INFP, D_INFP);
|
|
63 }
|
|
64 else
|
|
65 {
|
|
66 w = cpack((2\&.0 * sin(a) * cosh(b) / d), (-2\&.0 * cos(a) * sinh(b) / d));
|
|
67 }
|
|
68
|
|
69 return w;
|
|
70 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccsch (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Hyperbolic secant of a complex number\&. Calculated as in Open Office:
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
2\&.0 * sinh(a) * cos(b)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * a) - cos(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
-2\&.0 * cosh(2\&.0 * a) * sin(b)
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * a) - cos(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMCSCH_function
|
|
.PP
|
|
Definition at line 50 of file ccsch\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 complex w;
|
|
53 double a, b;
|
|
54 double d;
|
|
55
|
|
56 a = creal(z);
|
|
57 b = cimag(z);
|
|
58 d = cosh(2\&.0 * a) - cos(2\&.0 * b);
|
|
59 w = cpack((2\&.0 * sinh(a) * cos(b) / d), (-2\&.0 * cosh(a) * sin(b) / d));
|
|
60
|
|
61 return w;
|
|
62 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cdiv (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Division of two complex numbers\&.
|
|
.PP
|
|
Definition at line 173 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
174 {
|
|
175 complex w;
|
|
176 double a, b, c, d;
|
|
177 double q, v, x;
|
|
178
|
|
179 a = creal(y);
|
|
180 b = cimag(y);
|
|
181 c = creal(z);
|
|
182 d = cimag(z);
|
|
183
|
|
184 q = c * c + d * d;
|
|
185 v = a * c + b * d;
|
|
186 x = b * c - a * d;
|
|
187
|
|
188 w = cpack(v / q, x / q);
|
|
189 return w;
|
|
190 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cexp (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Returns e to the power of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 47 of file cexp\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cpack(), creal(), exp(), and sin()\&.
|
|
.PP
|
|
.nf
|
|
48 {
|
|
49 complex w;
|
|
50 double r, x, y;
|
|
51 x = creal(z);
|
|
52 y = cimag(z);
|
|
53 r = exp(x);
|
|
54 w = cpack(r * cos(y), r * sin(y));
|
|
55 return w;
|
|
56 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cfloor (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Floor value of complex number\&.
|
|
.PP
|
|
Definition at line 98 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
99 {
|
|
100 complex w;
|
|
101 w = cpack(floor(creal(z)), floor(cimag(z)));
|
|
102 return w;
|
|
103 }
|
|
.fi
|
|
.SS "double cimag (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Imaginary part of complex number\&.
|
|
.PP
|
|
Definition at line 50 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 return (IMAG_PART(z));
|
|
53 }
|
|
.fi
|
|
.SS "\fBcomplex\fP clog (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Natural logarithm of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 47 of file clog\&.c\&.
|
|
.PP
|
|
References atan2(), cabs(), cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
48 {
|
|
49 complex w;
|
|
50 double p, q;
|
|
51 p = log(cabs(z));
|
|
52 q = atan2(cimag(z), creal(z));
|
|
53 w = cpack(p, q);
|
|
54 return w;
|
|
55 }
|
|
.fi
|
|
.SS "\fBcomplex\fP clog10 (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Base 10 logarithmic value of complex number\&.
|
|
.PP
|
|
.nf
|
|
|
|
log z = log(z) / log(10)
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Complex_logarithm
|
|
.PP
|
|
Definition at line 42 of file clog10\&.c\&.
|
|
.PP
|
|
References cdiv(), clog(), and cpack()\&.
|
|
.PP
|
|
.nf
|
|
43 {
|
|
44 complex teen = cpack(10\&.0, 0\&.0);
|
|
45 complex w = cdiv(clog(z), clog(teen));
|
|
46 return w;
|
|
47 }
|
|
.fi
|
|
.SS "\fBcomplex\fP clogb (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Base 2 logarithmic value of complex number\&.
|
|
.PP
|
|
.nf
|
|
|
|
lb z = log(z) / log(2)
|
|
.fi
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Complex_logarithm
|
|
.PP
|
|
Definition at line 42 of file clogb\&.c\&.
|
|
.PP
|
|
References cdiv(), clog(), and cpack()\&.
|
|
.PP
|
|
.nf
|
|
43 {
|
|
44 complex two = cpack(2\&.0, 0\&.0);
|
|
45 complex w = cdiv(clog(z), clog(two));
|
|
46 return w;
|
|
47 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cmul (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Multiplication of two complex numbers\&.
|
|
.PP
|
|
Definition at line 153 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
154 {
|
|
155 complex w;
|
|
156 double a, b, c, d;
|
|
157
|
|
158 // (a+bi)(c+di)
|
|
159 a = creal(y);
|
|
160 b = cimag(y);
|
|
161 c = creal(z);
|
|
162 d = cimag(z);
|
|
163
|
|
164 // (ac -bd) + (ad + bc)i
|
|
165 w = cpack(a * c - b * d, a * d + b * c);
|
|
166 return w;
|
|
167 }
|
|
.fi
|
|
.SS "\fBcomplex\fP conj (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Definition at line 64 of file prim\&.c\&.
|
|
.PP
|
|
References cpack()\&.
|
|
.PP
|
|
.nf
|
|
65 {
|
|
66 IMAG_PART(z) = -IMAG_PART(z);
|
|
67 return cpack(REAL_PART(z), IMAG_PART(z));
|
|
68 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cpack (double x, double y)"
|
|
|
|
.PP
|
|
Pack two real numbers into a complex number\&.
|
|
.PP
|
|
Definition at line 74 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
75 {
|
|
76 complex z;
|
|
77
|
|
78 REAL_PART(z) = x;
|
|
79 IMAG_PART(z) = y;
|
|
80 return (z);
|
|
81 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cpow (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Complex number raised to a power\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 47 of file cpow\&.c\&.
|
|
.PP
|
|
References atan2(), cabs(), cimag(), cos(), cpack(), creal(), exp(), pow(), and sin()\&.
|
|
.PP
|
|
.nf
|
|
48 {
|
|
49 complex w;
|
|
50 double x, y, r, theta, absa, arga;
|
|
51
|
|
52 x = creal(z);
|
|
53 y = cimag(z);
|
|
54 absa = cabs(a);
|
|
55 if (absa == 0\&.0)
|
|
56 {
|
|
57 return cpack(0\&.0, + 0\&.0);
|
|
58 }
|
|
59 arga = atan2(cimag(a), creal(a));
|
|
60
|
|
61 r = pow(absa, x);
|
|
62 theta = x * arga;
|
|
63 if (y != 0\&.0)
|
|
64 {
|
|
65 r = r * exp(-y * arga);
|
|
66 theta = theta + y * log(absa);
|
|
67 }
|
|
68
|
|
69 w = cpack(r * cos(theta), r * sin(theta));
|
|
70 return w;
|
|
71 }
|
|
.fi
|
|
.SS "double creal (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Real part of complex number\&.
|
|
.PP
|
|
Definition at line 41 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
42 {
|
|
43 return (REAL_PART(z));
|
|
44 }
|
|
.fi
|
|
.SS "\fBcomplex\fP creci (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Reciprocal value of complex number\&.
|
|
.PP
|
|
Definition at line 196 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), conj(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
197 {
|
|
198 complex w;
|
|
199 double q, a, b;
|
|
200
|
|
201 a = creal(z);
|
|
202 b = cimag(conj(z));
|
|
203 q = a * a + b * b;
|
|
204 w = cpack(a / q, b / q);
|
|
205
|
|
206 return w;
|
|
207 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cround (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Division of two complex numbers\&.
|
|
.PP
|
|
Definition at line 120 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), creal(), and round()\&.
|
|
.PP
|
|
.nf
|
|
121 {
|
|
122 complex w;
|
|
123 w = cpack(round(creal(z)), round(cimag(z)));
|
|
124 return w;
|
|
125 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csec (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Secant of a complex number\&. Calculated as in Open Office:
|
|
.br
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMSEC_function
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
2\&.0 * cos(a) * cosh(b)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * b) + cos(2\&.0 * a)
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
2\&.0 * sin(a) * sinh(b)
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * b) + cos(2\&.0 * a)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 50 of file csec\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 complex w;
|
|
53 double a, b;
|
|
54 double d;
|
|
55
|
|
56 a = creal(z);
|
|
57 b = cimag(z);
|
|
58 d = cosh(2\&.0 * b) + cos(2\&.0 * a);
|
|
59
|
|
60 if (d == 0\&.0)
|
|
61 {
|
|
62 w = cpack(D_INFP, D_INFP);
|
|
63 }
|
|
64 else
|
|
65 {
|
|
66 w = cpack((2\&.0 * cos(a) * cosh(b) / d), (2\&.0 * sin(a) * sinh(b) / d));
|
|
67 }
|
|
68
|
|
69 return w;
|
|
70 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csech (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Hyperbolic secant of a complex number\&. Calculated as in Open Office:
|
|
.br
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMSECH_function
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
2\&.0 * cosh(a) * cos(b)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * a) + cos(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
.PP
|
|
.PP
|
|
.nf
|
|
-2\&.0 * sinh(2\&.0 * a) * sin(b)
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * a) + cos(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 50 of file csech\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 complex w;
|
|
53 double a, b;
|
|
54 double d;
|
|
55
|
|
56 a = creal(z);
|
|
57 b = cimag(z);
|
|
58 d = cosh(2\&.0 * a) + cos(2\&.0 * b);
|
|
59 w = cpack((2\&.0 * cosh(a) * cos(b) / d), (-2\&.0 * sinh(a) * sin(b) / d));
|
|
60
|
|
61 return w;
|
|
62 }
|
|
.fi
|
|
.SS "double csgn (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Complex signum\&. Mostly as specified in [IEEE Std 1003\&.1, 2013 Edition]: http://pubs.opengroup.org/onlinepubs/9699919799/basedefs/complex.h.html
|
|
.PP
|
|
More info is available at Wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Sign_function#Complex_signum
|
|
.PP
|
|
Definition at line 39 of file csgn\&.c\&.
|
|
.PP
|
|
References cimag(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
40 {
|
|
41 double a = creal(z);
|
|
42
|
|
43 if (a > 0\&.0)
|
|
44 {
|
|
45 return 1\&.0;
|
|
46 }
|
|
47 else if (a < 0\&.0)
|
|
48 {
|
|
49 return -1\&.0;
|
|
50 }
|
|
51 else
|
|
52 {
|
|
53 double b = cimag(z);
|
|
54 return b > 0\&.0 ? 1\&.0 : b < 0\&.0 ? -1\&.0 : 0\&.0;
|
|
55 }
|
|
56 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csin (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Sine of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
Calculated according to description at wikipedia:
|
|
.br
|
|
https://wikipedia.org/wiki/Sine#Sine_with_a_complex_argument
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
real = sin(a) * cosh(b)
|
|
imag = cos(a) * sinh(b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 54 of file csin\&.c\&.
|
|
.PP
|
|
References cchsh(), cimag(), cos(), cpack(), creal(), and sin()\&.
|
|
.PP
|
|
.nf
|
|
55 {
|
|
56 complex w;
|
|
57 double a, b;
|
|
58 double ch, sh;
|
|
59
|
|
60 a = creal(z);
|
|
61 b = cimag(z);
|
|
62 cchsh(b, &ch, &sh);
|
|
63 w = cpack((sin(a) * ch), (cos(a) * sh));
|
|
64
|
|
65 return w;
|
|
66 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csinh (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Hyperbolic sine of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
Calculated as in Open Office:
|
|
.br
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMSINH_function
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
real = sinh(a) * cos(b)
|
|
imag = cosh(a) * sin(b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 54 of file csinh\&.c\&.
|
|
.PP
|
|
References cchsh(), cimag(), cos(), cpack(), creal(), and sin()\&.
|
|
.PP
|
|
.nf
|
|
55 {
|
|
56 complex w;
|
|
57 double a, b;
|
|
58 double ch, sh;
|
|
59
|
|
60 a = creal(z);
|
|
61 b = cimag(z);
|
|
62 cchsh(a, &ch, &sh);
|
|
63 w = cpack(cos(b) * sh, sin(b) * ch);
|
|
64
|
|
65 return w;
|
|
66 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csqrt (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Square root of complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 47 of file csqrt\&.c\&.
|
|
.PP
|
|
References cabs(), cimag(), cpack(), creal(), and sqrt()\&.
|
|
.PP
|
|
.nf
|
|
48 {
|
|
49 complex w;
|
|
50 double x, y, r, t, scale;
|
|
51
|
|
52 x = creal(z);
|
|
53 y = cimag(z);
|
|
54
|
|
55 if (y == 0\&.0)
|
|
56 {
|
|
57 if (x == 0\&.0)
|
|
58 {
|
|
59 w = cpack(0\&.0, y);
|
|
60 }
|
|
61 else
|
|
62 {
|
|
63 r = fabs(x);
|
|
64 r = sqrt(r);
|
|
65 if (x < 0\&.0)
|
|
66 {
|
|
67 w = cpack(0\&.0, r);
|
|
68 }
|
|
69 else
|
|
70 {
|
|
71 w = cpack(r, y);
|
|
72 }
|
|
73 }
|
|
74 return w;
|
|
75 }
|
|
76 if (x == 0\&.0)
|
|
77 {
|
|
78 r = fabs(y);
|
|
79 r = sqrt(0\&.5 * r);
|
|
80 if (y > 0)
|
|
81 w = cpack(r, r);
|
|
82 else
|
|
83 w = cpack(r, -r);
|
|
84 return w;
|
|
85 }
|
|
86 /* Rescale to avoid internal overflow or underflow\&. */
|
|
87 if ((fabs(x) > 4\&.0) || (fabs(y) > 4\&.0))
|
|
88 {
|
|
89 x *= 0\&.25;
|
|
90 y *= 0\&.25;
|
|
91 scale = 2\&.0;
|
|
92 }
|
|
93 else
|
|
94 {
|
|
95 #if 1
|
|
96 x *= 1\&.8014398509481984e16; /* 2^54 */
|
|
97 y *= 1\&.8014398509481984e16;
|
|
98 scale = 7\&.450580596923828125e-9; /* 2^-27 */
|
|
99 #else
|
|
100 x *= 4\&.0;
|
|
101 y *= 4\&.0;
|
|
102 scale = 0\&.5;
|
|
103 #endif
|
|
104 }
|
|
105 w = cpack(x, y);
|
|
106 r = cabs(w);
|
|
107 if (x > 0)
|
|
108 {
|
|
109 t = sqrt(0\&.5 * r + 0\&.5 * x);
|
|
110 r = scale * fabs((0\&.5 * y) / t);
|
|
111 t *= scale;
|
|
112 }
|
|
113 else
|
|
114 {
|
|
115 r = sqrt(0\&.5 * r - 0\&.5 * x);
|
|
116 t = scale * fabs((0\&.5 * y) / r);
|
|
117 r *= scale;
|
|
118 }
|
|
119 if (y < 0)
|
|
120 w = cpack(t, -r);
|
|
121 else
|
|
122 w = cpack(t, r);
|
|
123 return w;
|
|
124 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csub (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Subtraction of two complex numbers\&.
|
|
.PP
|
|
Definition at line 142 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), and creal()\&.
|
|
.PP
|
|
.nf
|
|
143 {
|
|
144 complex w;
|
|
145 w = cpack(creal(y) - creal(z), cimag(y) - cimag(z));
|
|
146 return w;
|
|
147 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ctan (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Tangent of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
Calculated as in Open Office:
|
|
.br
|
|
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_IMTAN_function
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
sin(2\&.0 * a)
|
|
real = ------------------------------
|
|
cos(2\&.0 * a) + cosh(2\&.0 * b)
|
|
.PP
|
|
.nf
|
|
sinh(2.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
imag = ------------------------------
|
|
cos(2\&.0 * a) + cosh(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 59 of file ctan\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
60 {
|
|
61 complex w;
|
|
62 double a, b;
|
|
63 double d;
|
|
64
|
|
65 a = creal(z);
|
|
66 b = cimag(z);
|
|
67 d = cos(2\&.0 * a) + cosh(2\&.0 * b);
|
|
68
|
|
69 if (d == 0\&.0)
|
|
70 {
|
|
71 w = cpack(D_INFP, D_INFP);
|
|
72 }
|
|
73 else
|
|
74 {
|
|
75 w = cpack((sin(2\&.0 * a) / d), (sinh(2\&.0 * b) / d));
|
|
76 }
|
|
77
|
|
78 return w;
|
|
79 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ctanh (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Hyperbolic tangent of a complex number\&.
|
|
.PP
|
|
\fBVersion:\fP
|
|
.RS 4
|
|
1\&.1
|
|
.RE
|
|
.PP
|
|
\fBDate:\fP
|
|
.RS 4
|
|
2007/08/20
|
|
.RE
|
|
.PP
|
|
.PP
|
|
.nf
|
|
|
|
a+bi
|
|
sinh(2\&.0 * a)
|
|
real = ------------------------------
|
|
cosh(2\&.0 * a) + cos(2\&.0 * b)
|
|
.PP
|
|
.nf
|
|
sin(2.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
imag = ------------------------------
|
|
cosh(2\&.0 * a) + cos(2\&.0 * b)
|
|
.fi
|
|
.PP
|
|
|
|
.PP
|
|
Definition at line 57 of file ctanh\&.c\&.
|
|
.PP
|
|
References cimag(), cos(), cosh(), cpack(), creal(), sin(), and sinh()\&.
|
|
.PP
|
|
.nf
|
|
58 {
|
|
59 complex w;
|
|
60 double a, b;
|
|
61 double d;
|
|
62
|
|
63 a = creal(z);
|
|
64 b = cimag(z);
|
|
65 d = cosh(2\&.0 * a) + cos(2\&.0 * b);
|
|
66 w = cpack((sinh(2\&.0 * a) / d), (sin(2\&.0 * b) / d));
|
|
67
|
|
68 return w;
|
|
69 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ctrunc (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Truncated value of complex number\&.
|
|
.PP
|
|
Definition at line 87 of file prim\&.c\&.
|
|
.PP
|
|
References cimag(), cpack(), creal(), and trunc()\&.
|
|
.PP
|
|
.nf
|
|
88 {
|
|
89 complex w;
|
|
90 w = cpack(trunc(creal(z)), trunc(cimag(z)));
|
|
91 return w;
|
|
92 }
|
|
.fi
|
|
.SH "See also"
|
|
.PP
|
|
amath(1), amathc(3), amathr(3)
|