1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
|
// 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 {
if imag(x) == 0 && math.Abs(real(x)) <= 1 {
return complex(math.Asin(real(x)), imag(x))
}
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 {
if imag(x) == 0 && math.Abs(real(x)) <= 1 {
return complex(math.Asinh(real(x)), imag(x))
}
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 {
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 {
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
}
|
