1// Copyright 2010 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5package cmplx
6
7import "math"
8
9// The original C code, the long comment, and the constants
10// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
11// The go code is a simplified version of the original C.
12//
13// Cephes Math Library Release 2.8:  June, 2000
14// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
15//
16// The readme file at http://netlib.sandia.gov/cephes/ says:
17//    Some software in this archive may be from the book _Methods and
18// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
19// International, 1989) or from the Cephes Mathematical Library, a
20// commercial product. In either event, it is copyrighted by the author.
21// What you see here may be used freely but it comes with no support or
22// guarantee.
23//
24//   The two known misprints in the book are repaired here in the
25// source listings for the gamma function and the incomplete beta
26// integral.
27//
28//   Stephen L. Moshier
29//   [email protected]
30
31// Complex circular arc sine
32//
33// DESCRIPTION:
34//
35// Inverse complex sine:
36//                               2
37// w = -i clog( iz + csqrt( 1 - z ) ).
38//
39// casin(z) = -i casinh(iz)
40//
41// ACCURACY:
42//
43//                      Relative error:
44// arithmetic   domain     # trials      peak         rms
45//    DEC       -10,+10     10100       2.1e-15     3.4e-16
46//    IEEE      -10,+10     30000       2.2e-14     2.7e-15
47// Larger relative error can be observed for z near zero.
48// Also tested by csin(casin(z)) = z.
49
50// Asin returns the inverse sine of x.
51func Asin(x complex128) complex128 {
52	switch re, im := real(x), imag(x); {
53	case im == 0 && math.Abs(re) <= 1:
54		return complex(math.Asin(re), im)
55	case re == 0 && math.Abs(im) <= 1:
56		return complex(re, math.Asinh(im))
57	case math.IsNaN(im):
58		switch {
59		case re == 0:
60			return complex(re, math.NaN())
61		case math.IsInf(re, 0):
62			return complex(math.NaN(), re)
63		default:
64			return NaN()
65		}
66	case math.IsInf(im, 0):
67		switch {
68		case math.IsNaN(re):
69			return x
70		case math.IsInf(re, 0):
71			return complex(math.Copysign(math.Pi/4, re), im)
72		default:
73			return complex(math.Copysign(0, re), im)
74		}
75	case math.IsInf(re, 0):
76		return complex(math.Copysign(math.Pi/2, re), math.Copysign(re, im))
77	}
78	ct := complex(-imag(x), real(x)) // i * x
79	xx := x * x
80	x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
81	x2 := Sqrt(x1)                       // x2 = sqrt(1 - x*x)
82	w := Log(ct + x2)
83	return complex(imag(w), -real(w)) // -i * w
84}
85
86// Asinh returns the inverse hyperbolic sine of x.
87func Asinh(x complex128) complex128 {
88	switch re, im := real(x), imag(x); {
89	case im == 0 && math.Abs(re) <= 1:
90		return complex(math.Asinh(re), im)
91	case re == 0 && math.Abs(im) <= 1:
92		return complex(re, math.Asin(im))
93	case math.IsInf(re, 0):
94		switch {
95		case math.IsInf(im, 0):
96			return complex(re, math.Copysign(math.Pi/4, im))
97		case math.IsNaN(im):
98			return x
99		default:
100			return complex(re, math.Copysign(0.0, im))
101		}
102	case math.IsNaN(re):
103		switch {
104		case im == 0:
105			return x
106		case math.IsInf(im, 0):
107			return complex(im, re)
108		default:
109			return NaN()
110		}
111	case math.IsInf(im, 0):
112		return complex(math.Copysign(im, re), math.Copysign(math.Pi/2, im))
113	}
114	xx := x * x
115	x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
116	return Log(x + Sqrt(x1))            // log(x + sqrt(1 + x*x))
117}
118
119// Complex circular arc cosine
120//
121// DESCRIPTION:
122//
123// w = arccos z  =  PI/2 - arcsin z.
124//
125// ACCURACY:
126//
127//                      Relative error:
128// arithmetic   domain     # trials      peak         rms
129//    DEC       -10,+10      5200      1.6e-15      2.8e-16
130//    IEEE      -10,+10     30000      1.8e-14      2.2e-15
131
132// Acos returns the inverse cosine of x.
133func Acos(x complex128) complex128 {
134	w := Asin(x)
135	return complex(math.Pi/2-real(w), -imag(w))
136}
137
138// Acosh returns the inverse hyperbolic cosine of x.
139func Acosh(x complex128) complex128 {
140	if x == 0 {
141		return complex(0, math.Copysign(math.Pi/2, imag(x)))
142	}
143	w := Acos(x)
144	if imag(w) <= 0 {
145		return complex(-imag(w), real(w)) // i * w
146	}
147	return complex(imag(w), -real(w)) // -i * w
148}
149
150// Complex circular arc tangent
151//
152// DESCRIPTION:
153//
154// If
155//     z = x + iy,
156//
157// then
158//          1       (    2x     )
159// Re w  =  - arctan(-----------)  +  k PI
160//          2       (     2    2)
161//                  (1 - x  - y )
162//
163//               ( 2         2)
164//          1    (x  +  (y+1) )
165// Im w  =  - log(------------)
166//          4    ( 2         2)
167//               (x  +  (y-1) )
168//
169// Where k is an arbitrary integer.
170//
171// catan(z) = -i catanh(iz).
172//
173// ACCURACY:
174//
175//                      Relative error:
176// arithmetic   domain     # trials      peak         rms
177//    DEC       -10,+10      5900       1.3e-16     7.8e-18
178//    IEEE      -10,+10     30000       2.3e-15     8.5e-17
179// The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
180// had peak relative error 1.5e-16, rms relative error
181// 2.9e-17.  See also clog().
182
183// Atan returns the inverse tangent of x.
184func Atan(x complex128) complex128 {
185	switch re, im := real(x), imag(x); {
186	case im == 0:
187		return complex(math.Atan(re), im)
188	case re == 0 && math.Abs(im) <= 1:
189		return complex(re, math.Atanh(im))
190	case math.IsInf(im, 0) || math.IsInf(re, 0):
191		if math.IsNaN(re) {
192			return complex(math.NaN(), math.Copysign(0, im))
193		}
194		return complex(math.Copysign(math.Pi/2, re), math.Copysign(0, im))
195	case math.IsNaN(re) || math.IsNaN(im):
196		return NaN()
197	}
198	x2 := real(x) * real(x)
199	a := 1 - x2 - imag(x)*imag(x)
200	if a == 0 {
201		return NaN()
202	}
203	t := 0.5 * math.Atan2(2*real(x), a)
204	w := reducePi(t)
205
206	t = imag(x) - 1
207	b := x2 + t*t
208	if b == 0 {
209		return NaN()
210	}
211	t = imag(x) + 1
212	c := (x2 + t*t) / b
213	return complex(w, 0.25*math.Log(c))
214}
215
216// Atanh returns the inverse hyperbolic tangent of x.
217func Atanh(x complex128) complex128 {
218	z := complex(-imag(x), real(x)) // z = i * x
219	z = Atan(z)
220	return complex(imag(z), -real(z)) // z = -i * z
221}
222