mirror of https://gitlab.com/rnger/amath
1752 lines
33 KiB
Groff
1752 lines
33 KiB
Groff
.\" Copyright (c) 2014-2021 Carsten Sonne Larsen <cs@innolan.net>
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.\" All rights reserved.
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.\"
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.\" Redistribution and use in source and binary forms, with or without
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.\" modification, are permitted provided that the following conditions
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.\" are met:
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.\" 1. Redistributions of source code must retain the above copyright
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.\" notice, this list of conditions and the following disclaimer.
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.\" 2. Redistributions in binary form must reproduce the above copyright
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.\" notice, this list of conditions and the following disclaimer in the
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.\" documentation and/or other materials provided with the distribution.
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.\"
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.\" THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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.\" IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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.\" OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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.\" IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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.\" INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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.\" NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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.\" DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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.\" THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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.\" (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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.\" THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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.TH "mathi.h" 3 "Version 1.8.5" "August 07 2018"
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.SH NAME
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mathi.h \- Complex numbers math library
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.SH SYNOPSIS
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.br
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.SS "Data Structures"
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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 \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 \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 \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 \fBconj\fP (\fBcomplex\fP z)"
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.br
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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 \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 \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 \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 \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 for handling complex numbers\&.
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Mostly as specified in IEEE Std 1003\&.1, 2013 Edition:
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.br
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http://pubs.opengroup.org/onlinepubs/9699919799/basedefs/complex.h.html
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.PP
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Definition in file \fBmathi\&.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 57 of file prim\&.c\&.
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.PP
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.nf
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58 {
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59 return hypot(creal(z), cimag(z));
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60 }
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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 44 of file cacos\&.c\&.
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.PP
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.nf
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45 {
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46 complex a = cpack(1\&.0, 0\&.0);
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47 complex i = cpack(0\&.0, 1\&.0);
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48 complex j = cpack(0\&.0, -1\&.0);
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49 complex p = csub(a, cmul(z, z));
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50 complex q = clog(cadd(z, cmul(i, csqrt(p))));
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51 complex w = cmul(j, q);
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52 return w;
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53 }
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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 44 of file cacosh\&.c\&.
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.PP
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.nf
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45 {
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46 complex one = cpack(1\&.0, 0\&.0);
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47 complex a = csub(cmul(z, z), one);
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48 complex b = cadd(z, csqrt(a));
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49 complex w = clog(b);
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50 return w;
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51 }
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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 44 of file cacot\&.c\&.
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.PP
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.nf
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45 {
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46 complex one = cpack(1\&.0, 0\&.0);
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47 complex two = cpack(2\&.0, 0\&.0);
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48 complex i = cpack(0\&.0, 1\&.0);
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49 complex iz = cdiv(i, z);
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50 complex p = clog(csub(one, iz));
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51 complex q = clog(cadd(one, iz));
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52 complex w = cmul(cdiv(i, two), csub(p, q));
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53 return w;
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54 }
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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 44 of file cacoth\&.c\&.
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.PP
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.nf
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45 {
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46 complex half = cpack(0\&.5, 0\&.0);
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47 complex one = cpack(1\&.0, 0\&.0);
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48 complex a = clog(cadd(z, one));
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49 complex b = clog(csub(z, one));
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50 complex c = csub(a, b);
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51 complex w = cmul(half, c);
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52 return w;
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53 }
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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 44 of file cacsc\&.c\&.
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.PP
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.nf
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45 {
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46 complex one = cpack(1\&.0, 0\&.0);
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47 complex i = cpack(0\&.0, 1\&.0);
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48 complex j = cpack(0\&.0, -1\&.0);
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49 complex iz = cdiv(i, z);
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50 complex z2 = cmul(z, z);
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51 complex p = cdiv(one, z2);
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52 complex q = csqrt(csub(one, p));
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53 complex w = cmul(j, clog(cadd(q, iz)));
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54 return w;
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55 }
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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
|
|
.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 44 of file cacsch\&.c\&.
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.PP
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.nf
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45 {
|
|
46 complex one = cpack(1\&.0, 0\&.0);
|
|
47 complex a = creci(cmul(z, z));
|
|
48 complex b = csqrt(cadd(one, a));
|
|
49 complex c = cadd(b, creci(z));
|
|
50 complex w = clog(c);
|
|
51 return w;
|
|
52 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cadd (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
.PP
|
|
Addition of two complex numbers\&.
|
|
.PP
|
|
Definition at line 129 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
130 {
|
|
131 complex w;
|
|
132 w = cpack(creal(y) + creal(z), cimag(y) + cimag(z));
|
|
133 return w;
|
|
134 }
|
|
.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 44 of file casec\&.c\&.
|
|
.PP
|
|
.nf
|
|
45 {
|
|
46 complex one = cpack(1\&.0, 0\&.0);
|
|
47 complex i = cpack(0\&.0, 1\&.0);
|
|
48 complex j = cpack(0\&.0, -1\&.0);
|
|
49 complex rz = creci(z);
|
|
50 complex z2 = cmul(z, z);
|
|
51 complex p = cdiv(one, z2);
|
|
52 complex q = csqrt(csub(one, p));
|
|
53 complex w = cmul(j, clog(cadd(cmul(i, q), rz)));
|
|
54 return w;
|
|
55 }
|
|
.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 45 of file casech\&.c\&.
|
|
.PP
|
|
.nf
|
|
46 {
|
|
47 complex one = cpack(1\&.0, 0\&.0);
|
|
48 complex a = creci(cmul(z, z));
|
|
49 complex b = csqrt(csub(a, one));
|
|
50 complex c = cadd(b, creci(z));
|
|
51 complex w = clog(c);
|
|
52 return w;
|
|
53 }
|
|
.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 44 of file casin\&.c\&.
|
|
.PP
|
|
.nf
|
|
45 {
|
|
46 complex one = cpack(1\&.0, 0\&.0);
|
|
47 complex i = cpack(0\&.0, 1\&.0);
|
|
48 complex j = cpack(0\&.0, -1\&.0);
|
|
49 complex iz = cmul(i, z);
|
|
50 complex z2 = cmul(z, z);
|
|
51 complex p = csqrt(csub(one, z2));
|
|
52 complex q = clog(cadd(iz, p));
|
|
53 complex w = cmul(j, q);
|
|
54 return w;
|
|
55 }
|
|
.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 49 of file casinh\&.c\&.
|
|
.PP
|
|
.nf
|
|
50 {
|
|
51 complex one = cpack(1\&.0, 0\&.0);
|
|
52 complex a = cadd(cmul(z, z), one);
|
|
53 complex b = cadd(z, csqrt(a));
|
|
54 complex w = clog(b);
|
|
55 return w;
|
|
56 }
|
|
.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 44 of file catan\&.c\&.
|
|
.PP
|
|
.nf
|
|
45 {
|
|
46 complex one = cpack(1\&.0, 0\&.0);
|
|
47 complex two = cpack(2\&.0, 0\&.0);
|
|
48 complex i = cpack(0\&.0, 1\&.0);
|
|
49 complex iz = cmul(i, z);
|
|
50 complex p = clog(csub(one, iz));
|
|
51 complex q = clog(cadd(one, iz));
|
|
52 complex w = cmul(cdiv(i, two), csub(p, q));
|
|
53 return w;
|
|
54 }
|
|
.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 44 of file catanh\&.c\&.
|
|
.PP
|
|
.nf
|
|
45 {
|
|
46 complex half = cpack(0\&.5, 0\&.0);
|
|
47 complex one = cpack(1\&.0, 0\&.0);
|
|
48 complex a = clog(cadd(one, z));
|
|
49 complex b = clog(csub(one, z));
|
|
50 complex c = csub(a, b);
|
|
51 complex w = cmul(half, c);
|
|
52 return w;
|
|
53 }
|
|
.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 41 of file ccbrt\&.c\&.
|
|
.PP
|
|
.nf
|
|
42 {
|
|
43 complex onethird = cpack(1\&.0 / 3\&.0, 0\&.0);
|
|
44 complex a = cmul(onethird, clog(z));
|
|
45 complex w = cexp(a);
|
|
46 return w;
|
|
47 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cceil (\fBcomplex\fP z)"
|
|
.PP
|
|
Ceiling value of complex number\&.
|
|
.PP
|
|
Definition at line 107 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
108 {
|
|
109 complex w;
|
|
110 w = cpack(ceil(creal(z)), ceil(cimag(z)));
|
|
111 return w;
|
|
112 }
|
|
.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 47 of file ccos\&.c\&.
|
|
.PP
|
|
.nf
|
|
48 {
|
|
49 complex w;
|
|
50 double a, b;
|
|
51 double ch, sh;
|
|
52
|
|
53 a = creal(z);
|
|
54 b = cimag(z);
|
|
55 cchsh(b, &ch, &sh);
|
|
56 w = cpack((cos(a) * ch), (-sin(a) * sh));
|
|
57
|
|
58 return w;
|
|
59 }
|
|
.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
|
|
.nf
|
|
a+bi
|
|
real = cosh(a) * cos(b)
|
|
imag = sinh(a) * sin(b)
|
|
.fi
|
|
.PP
|
|
Definition at line 50 of file ccosh\&.c\&.
|
|
.PP
|
|
.nf
|
|
51 {
|
|
52 complex w;
|
|
53 double a, b;
|
|
54 double ch, sh;
|
|
55
|
|
56 a = creal(z);
|
|
57 b = cimag(z);
|
|
58 cchsh(a, &ch, &sh);
|
|
59 w = cpack(cos(b) * ch, sin(b) * sh);
|
|
60
|
|
61 return w;
|
|
62 }
|
|
.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
|
|
.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 48 of file ccot\&.c\&.
|
|
.PP
|
|
.nf
|
|
49 {
|
|
50 complex w;
|
|
51 double a, b;
|
|
52 double d;
|
|
53
|
|
54 a = creal(z);
|
|
55 b = cimag(z);
|
|
56 d = cosh(2\&.0 * b) - cos(2\&.0 * a);
|
|
57
|
|
58 if (d == 0\&.0)
|
|
59 {
|
|
60 w = cpack((double)INFP, (double)INFP);
|
|
61 }
|
|
62 else
|
|
63 {
|
|
64 w = cpack((sin(2\&.0 * a) / d), (-sinh(2\&.0 * b) / d));
|
|
65 }
|
|
66
|
|
67 return w;
|
|
68 }
|
|
.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 50 of file ccoth\&.c\&.
|
|
.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(sinh(2\&.0 * a) / d, -sin(2\&.0 * b) / d);
|
|
60
|
|
61 return w;
|
|
62 }
|
|
.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 48 of file ccsc\&.c\&.
|
|
.PP
|
|
.nf
|
|
49 {
|
|
50 complex w;
|
|
51 double a, b;
|
|
52 double d;
|
|
53
|
|
54 a = creal(z);
|
|
55 b = cimag(z);
|
|
56 d = cosh(2\&.0 * b) - cos(2\&.0 * a);
|
|
57
|
|
58 if (d == 0\&.0)
|
|
59 {
|
|
60 w = cpack((double)INFP, (double)INFP);
|
|
61 }
|
|
62 else
|
|
63 {
|
|
64 w = cpack((2\&.0 * sin(a) * cosh(b) / d), (-2\&.0 * cos(a) * sinh(b) / d));
|
|
65 }
|
|
66
|
|
67 return w;
|
|
68 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ccsch (\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:_IMCSCH_function
|
|
.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
|
|
Definition at line 48 of file ccsch\&.c\&.
|
|
.PP
|
|
.nf
|
|
49 {
|
|
50 complex w;
|
|
51 double a, b;
|
|
52 double d;
|
|
53
|
|
54 a = creal(z);
|
|
55 b = cimag(z);
|
|
56 d = cosh(2\&.0 * a) - cos(2\&.0 * b);
|
|
57 w = cpack((2\&.0 * sinh(a) * cos(b) / d), (-2\&.0 * cosh(a) * sin(b) / d));
|
|
58
|
|
59 return w;
|
|
60 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cdiv (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Division of two complex numbers\&.
|
|
.PP
|
|
Definition at line 171 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
172 {
|
|
173 complex w;
|
|
174 double a, b, c, d;
|
|
175 double q, v, x;
|
|
176
|
|
177 a = creal(y);
|
|
178 b = cimag(y);
|
|
179 c = creal(z);
|
|
180 d = cimag(z);
|
|
181
|
|
182 q = c * c + d * d;
|
|
183 v = a * c + b * d;
|
|
184 x = b * c - a * d;
|
|
185
|
|
186 w = cpack(v / q, x / q);
|
|
187 return w;
|
|
188 }
|
|
.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 45 of file cexp\&.c\&.
|
|
.PP
|
|
.nf
|
|
46 {
|
|
47 complex w;
|
|
48 double r, x, y;
|
|
49 x = creal(z);
|
|
50 y = cimag(z);
|
|
51 r = exp(x);
|
|
52 w = cpack(r * cos(y), r * sin(y));
|
|
53 return w;
|
|
54 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cfloor (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Floor value of complex number\&.
|
|
.PP
|
|
Definition at line 96 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
97 {
|
|
98 complex w;
|
|
99 w = cpack(floor(creal(z)), floor(cimag(z)));
|
|
100 return w;
|
|
101 }
|
|
.fi
|
|
.SS "double cimag (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Imaginary part of complex number\&.
|
|
.PP
|
|
Definition at line 48 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
49 {
|
|
50 return (IMAG_PART(z));
|
|
51 }
|
|
.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 45 of file clog\&.c\&.
|
|
.PP
|
|
.nf
|
|
46 {
|
|
47 complex w;
|
|
48 double p, q;
|
|
49 p = log(cabs(z));
|
|
50 q = atan2(cimag(z), creal(z));
|
|
51 w = cpack(p, q);
|
|
52 return w;
|
|
53 }
|
|
.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 41 of file clog10\&.c\&.
|
|
.PP
|
|
.nf
|
|
42 {
|
|
43 complex teen = cpack(10\&.0, 0\&.0);
|
|
44 complex w = cdiv(clog(z), clog(teen));
|
|
45 return w;
|
|
46 }
|
|
.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 41 of file clogb\&.c\&.
|
|
.PP
|
|
.nf
|
|
42 {
|
|
43 complex two = cpack(2\&.0, 0\&.0);
|
|
44 complex w = cdiv(clog(z), clog(two));
|
|
45 return w;
|
|
46 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cmul (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Multiplication of two complex numbers\&.
|
|
.PP
|
|
Definition at line 151 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
152 {
|
|
153 complex w;
|
|
154 double a, b, c, d;
|
|
155
|
|
156 // (a+bi)(c+di)
|
|
157 a = creal(y);
|
|
158 b = cimag(y);
|
|
159 c = creal(z);
|
|
160 d = cimag(z);
|
|
161
|
|
162 // (ac -bd) + (ad + bc)i
|
|
163 w = cpack(a * c - b * d, a * d + b * c);
|
|
164 return w;
|
|
165 }
|
|
.fi
|
|
.SS "\fBcomplex\fP conj (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Definition at line 62 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
63 {
|
|
64 IMAG_PART(z) = -IMAG_PART(z);
|
|
65 return cpack(REAL_PART(z), IMAG_PART(z));
|
|
66 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cpack (double x, double y)"
|
|
|
|
.PP
|
|
Pack two real numbers into a complex number\&.
|
|
.PP
|
|
Definition at line 72 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
73 {
|
|
74 complex z;
|
|
75
|
|
76 REAL_PART(z) = x;
|
|
77 IMAG_PART(z) = y;
|
|
78 return (z);
|
|
79 }
|
|
.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 45 of file cpow\&.c\&.
|
|
.PP
|
|
.nf
|
|
46 {
|
|
47 complex w;
|
|
48 double x, y, r, theta, absa, arga;
|
|
49
|
|
50 x = creal(z);
|
|
51 y = cimag(z);
|
|
52 absa = cabs(a);
|
|
53 if (absa == 0\&.0)
|
|
54 {
|
|
55 return cpack(0\&.0, + 0\&.0);
|
|
56 }
|
|
57 arga = atan2(cimag(a), creal(a));
|
|
58
|
|
59 r = pow(absa, x);
|
|
60 theta = x * arga;
|
|
61 if (y != 0\&.0)
|
|
62 {
|
|
63 r = r * exp(-y * arga);
|
|
64 theta = theta + y * log(absa);
|
|
65 }
|
|
66
|
|
67 w = cpack(r * cos(theta), r * sin(theta));
|
|
68 return w;
|
|
69 }
|
|
.fi
|
|
.SS "double creal (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Real part of complex number\&.
|
|
.PP
|
|
Definition at line 39 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
40 {
|
|
41 return (REAL_PART(z));
|
|
42 }
|
|
.fi
|
|
.SS "\fBcomplex\fP creci (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Reciprocal value of complex number\&.
|
|
.PP
|
|
Definition at line 194 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
195 {
|
|
196 complex w;
|
|
197 double q, a, b;
|
|
198
|
|
199 a = creal(z);
|
|
200 b = cimag(conj(z));
|
|
201 q = a * a + b * b;
|
|
202 w = cpack(a / q, b / q);
|
|
203
|
|
204 return w;
|
|
205 }
|
|
.fi
|
|
.SS "\fBcomplex\fP cround (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Division of two complex numbers\&.
|
|
.PP
|
|
Definition at line 118 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
119 {
|
|
120 complex w;
|
|
121 w = cpack(round(creal(z)), round(cimag(z)));
|
|
122 return w;
|
|
123 }
|
|
.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 48 of file csec\&.c\&.
|
|
.PP
|
|
.nf
|
|
49 {
|
|
50 complex w;
|
|
51 double a, b;
|
|
52 double d;
|
|
53
|
|
54 a = creal(z);
|
|
55 b = cimag(z);
|
|
56 d = cosh(2\&.0 * b) + cos(2\&.0 * a);
|
|
57
|
|
58 if (d == 0\&.0)
|
|
59 {
|
|
60 w = cpack((double)INFP, (double)INFP);
|
|
61 }
|
|
62 else
|
|
63 {
|
|
64 w = cpack((2\&.0 * cos(a) * cosh(b) / d), (2\&.0 * sin(a) * sinh(b) / d));
|
|
65 }
|
|
66
|
|
67 return w;
|
|
68 }
|
|
.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 48 of file csech\&.c\&.
|
|
.PP
|
|
.nf
|
|
49 {
|
|
50 complex w;
|
|
51 double a, b;
|
|
52 double d;
|
|
53
|
|
54 a = creal(z);
|
|
55 b = cimag(z);
|
|
56 d = cosh(2\&.0 * a) + cos(2\&.0 * b);
|
|
57 w = cpack((2\&.0 * cosh(a) * cos(b) / d), (-2\&.0 * sinh(a) * sin(b) / d));
|
|
58
|
|
59 return w;
|
|
60 }
|
|
.fi
|
|
.SS "double csgn (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Complex signum\&. 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
|
|
.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 52 of file csin\&.c\&.
|
|
.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(b, &ch, &sh);
|
|
61 w = cpack((sin(a) * ch), (cos(a) * sh));
|
|
62
|
|
63 return w;
|
|
64 }
|
|
.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 52 of file csinh\&.c\&.
|
|
.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) * sh, sin(b) * ch);
|
|
62
|
|
63 return w;
|
|
64 }
|
|
.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 45 of file csqrt\&.c\&.
|
|
.PP
|
|
.nf
|
|
46 {
|
|
47 complex w;
|
|
48 double x, y, r, t, scale;
|
|
49
|
|
50 x = creal(z);
|
|
51 y = cimag(z);
|
|
52
|
|
53 if (y == 0\&.0)
|
|
54 {
|
|
55 if (x == 0\&.0)
|
|
56 {
|
|
57 w = cpack(0\&.0, y);
|
|
58 }
|
|
59 else
|
|
60 {
|
|
61 r = fabs(x);
|
|
62 r = sqrt(r);
|
|
63 if (x < 0\&.0)
|
|
64 {
|
|
65 w = cpack(0\&.0, r);
|
|
66 }
|
|
67 else
|
|
68 {
|
|
69 w = cpack(r, y);
|
|
70 }
|
|
71 }
|
|
72 return w;
|
|
73 }
|
|
74 if (x == 0\&.0)
|
|
75 {
|
|
76 r = fabs(y);
|
|
77 r = sqrt(0\&.5 * r);
|
|
78 if (y > 0)
|
|
79 w = cpack(r, r);
|
|
80 else
|
|
81 w = cpack(r, -r);
|
|
82 return w;
|
|
83 }
|
|
84 /* Rescale to avoid internal overflow or underflow\&. */
|
|
85 if ((fabs(x) > 4\&.0) || (fabs(y) > 4\&.0))
|
|
86 {
|
|
87 x *= 0\&.25;
|
|
88 y *= 0\&.25;
|
|
89 scale = 2\&.0;
|
|
90 }
|
|
91 else
|
|
92 {
|
|
93 #if 1
|
|
94 x *= 1\&.8014398509481984e16; /* 2^54 */
|
|
95 y *= 1\&.8014398509481984e16;
|
|
96 scale = 7\&.450580596923828125e-9; /* 2^-27 */
|
|
97 #else
|
|
98 x *= 4\&.0;
|
|
99 y *= 4\&.0;
|
|
100 scale = 0\&.5;
|
|
101 #endif
|
|
102 }
|
|
103 w = cpack(x, y);
|
|
104 r = cabs(w);
|
|
105 if (x > 0)
|
|
106 {
|
|
107 t = sqrt(0\&.5 * r + 0\&.5 * x);
|
|
108 r = scale * fabs((0\&.5 * y) / t);
|
|
109 t *= scale;
|
|
110 }
|
|
111 else
|
|
112 {
|
|
113 r = sqrt(0\&.5 * r - 0\&.5 * x);
|
|
114 t = scale * fabs((0\&.5 * y) / r);
|
|
115 r *= scale;
|
|
116 }
|
|
117 if (y < 0)
|
|
118 w = cpack(t, -r);
|
|
119 else
|
|
120 w = cpack(t, r);
|
|
121 return w;
|
|
122 }
|
|
.fi
|
|
.SS "\fBcomplex\fP csub (\fBcomplex\fP a, \fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Subtraction of two complex numbers\&.
|
|
.PP
|
|
Definition at line 140 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
141 {
|
|
142 complex w;
|
|
143 w = cpack(creal(y) - creal(z), cimag(y) - cimag(z));
|
|
144 return w;
|
|
145 }
|
|
.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 57 of file ctan\&.c\&.
|
|
.PP
|
|
.nf
|
|
58 {
|
|
59 complex w;
|
|
60 double a, b;
|
|
61 double d;
|
|
62
|
|
63 a = creal(z);
|
|
64 b = cimag(z);
|
|
65 d = cos(2\&.0 * a) + cosh(2\&.0 * b);
|
|
66
|
|
67 if (d == 0\&.0)
|
|
68 {
|
|
69 w = cpack((double)INFP, (double)INFP);
|
|
70 }
|
|
71 else
|
|
72 {
|
|
73 w = cpack((sin(2\&.0 * a) / d), (sinh(2\&.0 * b) / d));
|
|
74 }
|
|
75
|
|
76 return w;
|
|
77 }
|
|
.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 55 of file ctanh\&.c\&.
|
|
.PP
|
|
.nf
|
|
56 {
|
|
57 complex w;
|
|
58 double a, b;
|
|
59 double d;
|
|
60
|
|
61 a = creal(z);
|
|
62 b = cimag(z);
|
|
63 d = cosh(2\&.0 * a) + cos(2\&.0 * b);
|
|
64 w = cpack((sinh(2\&.0 * a) / d), (sin(2\&.0 * b) / d));
|
|
65
|
|
66 return w;
|
|
67 }
|
|
.fi
|
|
.SS "\fBcomplex\fP ctrunc (\fBcomplex\fP z)"
|
|
|
|
.PP
|
|
Truncated value of complex number\&.
|
|
.PP
|
|
Definition at line 85 of file prim\&.c\&.
|
|
.PP
|
|
.nf
|
|
86 {
|
|
87 complex w;
|
|
88 w = cpack(trunc(creal(z)), trunc(cimag(z)));
|
|
89 return w;
|
|
90 }
|
|
.fi
|
|
.SH HOMEPAGE
|
|
https://amath.innolan.net/
|
|
.SH AUTHORS
|
|
.PP
|
|
Written by Carsten Sonne Larsen <cs@innolan.net>. Some code in the
|
|
library is derived from software written by Stephen L. Moshier.
|
|
.SH COPYRIGHT
|
|
Copyright (c) 2014-2021 Carsten Sonne Larsen <cs@innolan.net>
|
|
.br
|
|
Copyright (c) 2007 The NetBSD Foundation, Inc.
|
|
.SH "See also"
|
|
amath(1), amathc(3), amathr(3) |