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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmplx
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex circular arc sine
//
// DESCRIPTION:
//
// Inverse complex sine:
// 2
// w = -i clog( iz + csqrt( 1 - z ) ).
//
// casin(z) = -i casinh(iz)
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 10100 2.1e-15 3.4e-16
// IEEE -10,+10 30000 2.2e-14 2.7e-15
// Larger relative error can be observed for z near zero.
// Also tested by csin(casin(z)) = z.
// Asin returns the inverse sine of x.
func Asin(x complex128) complex128 {
switch re, im := real(x), imag(x); {
case im == 0 && math.Abs(re) <= 1:
return complex(math.Asin(re), im)
case re == 0 && math.Abs(im) <= 1:
return complex(re, math.Asinh(im))
case math.IsNaN(im):
switch {
case re == 0:
return complex(re, math.NaN())
case math.IsInf(re, 0):
return complex(math.NaN(), re)
default:
return NaN()
}
case math.IsInf(im, 0):
switch {
case math.IsNaN(re):
return x
case math.IsInf(re, 0):
return complex(math.Copysign(math.Pi/4, re), im)
default:
return complex(math.Copysign(0, re), im)
}
case math.IsInf(re, 0):
return complex(math.Copysign(math.Pi/2, re), math.Copysign(re, im))
}
ct := complex(-imag(x), real(x)) // i * x
xx := x * x
x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
x2 := Sqrt(x1) // x2 = sqrt(1 - x*x)
w := Log(ct + x2)
return complex(imag(w), -real(w)) // -i * w
}
// Asinh returns the inverse hyperbolic sine of x.
func Asinh(x complex128) complex128 {
switch re, im := real(x), imag(x); {
case im == 0 && math.Abs(re) <= 1:
return complex(math.Asinh(re), im)
case re == 0 && math.Abs(im) <= 1:
return complex(re, math.Asin(im))
case math.IsInf(re, 0):
switch {
case math.IsInf(im, 0):
return complex(re, math.Copysign(math.Pi/4, im))
case math.IsNaN(im):
return x
default:
return complex(re, math.Copysign(0.0, im))
}
case math.IsNaN(re):
switch {
case im == 0:
return x
case math.IsInf(im, 0):
return complex(im, re)
default:
return NaN()
}
case math.IsInf(im, 0):
return complex(math.Copysign(im, re), math.Copysign(math.Pi/2, im))
}
xx := x * x
x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
return Log(x + Sqrt(x1)) // log(x + sqrt(1 + x*x))
}
// Complex circular arc cosine
//
// DESCRIPTION:
//
// w = arccos z = PI/2 - arcsin z.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 5200 1.6e-15 2.8e-16
// IEEE -10,+10 30000 1.8e-14 2.2e-15
// Acos returns the inverse cosine of x.
func Acos(x complex128) complex128 {
w := Asin(x)
return complex(math.Pi/2-real(w), -imag(w))
}
// Acosh returns the inverse hyperbolic cosine of x.
func Acosh(x complex128) complex128 {
if x == 0 {
return complex(0, math.Copysign(math.Pi/2, imag(x)))
}
w := Acos(x)
if imag(w) <= 0 {
return complex(-imag(w), real(w)) // i * w
}
return complex(imag(w), -real(w)) // -i * w
}
// Complex circular arc tangent
//
// DESCRIPTION:
//
// If
// z = x + iy,
//
// then
// 1 ( 2x )
// Re w = - arctan(-----------) + k PI
// 2 ( 2 2)
// (1 - x - y )
//
// ( 2 2)
// 1 (x + (y+1) )
// Im w = - log(------------)
// 4 ( 2 2)
// (x + (y-1) )
//
// Where k is an arbitrary integer.
//
// catan(z) = -i catanh(iz).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 5900 1.3e-16 7.8e-18
// IEEE -10,+10 30000 2.3e-15 8.5e-17
// The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
// had peak relative error 1.5e-16, rms relative error
// 2.9e-17. See also clog().
// Atan returns the inverse tangent of x.
func Atan(x complex128) complex128 {
switch re, im := real(x), imag(x); {
case im == 0:
return complex(math.Atan(re), im)
case re == 0 && math.Abs(im) <= 1:
return complex(re, math.Atanh(im))
case math.IsInf(im, 0) || math.IsInf(re, 0):
if math.IsNaN(re) {
return complex(math.NaN(), math.Copysign(0, im))
}
return complex(math.Copysign(math.Pi/2, re), math.Copysign(0, im))
case math.IsNaN(re) || math.IsNaN(im):
return NaN()
}
x2 := real(x) * real(x)
a := 1 - x2 - imag(x)*imag(x)
if a == 0 {
return NaN()
}
t := 0.5 * math.Atan2(2*real(x), a)
w := reducePi(t)
t = imag(x) - 1
b := x2 + t*t
if b == 0 {
return NaN()
}
t = imag(x) + 1
c := (x2 + t*t) / b
return complex(w, 0.25*math.Log(c))
}
// Atanh returns the inverse hyperbolic tangent of x.
func Atanh(x complex128) complex128 {
z := complex(-imag(x), real(x)) // z = i * x
z = Atan(z)
return complex(imag(z), -real(z)) // z = -i * z
}
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