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Source file src/math/cmplx/asin.go

Documentation: math/cmplx

     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  
     5  package cmplx
     6  
     7  import "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  //   moshier@na-net.ornl.gov
    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.
    51  func Asin(x complex128) complex128 {
    52  	if imag(x) == 0 && math.Abs(real(x)) <= 1 {
    53  		return complex(math.Asin(real(x)), imag(x))
    54  	}
    55  	ct := complex(-imag(x), real(x)) // i * x
    56  	xx := x * x
    57  	x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
    58  	x2 := Sqrt(x1)                       // x2 = sqrt(1 - x*x)
    59  	w := Log(ct + x2)
    60  	return complex(imag(w), -real(w)) // -i * w
    61  }
    62  
    63  // Asinh returns the inverse hyperbolic sine of x.
    64  func Asinh(x complex128) complex128 {
    65  	if imag(x) == 0 && math.Abs(real(x)) <= 1 {
    66  		return complex(math.Asinh(real(x)), imag(x))
    67  	}
    68  	xx := x * x
    69  	x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
    70  	return Log(x + Sqrt(x1))            // log(x + sqrt(1 + x*x))
    71  }
    72  
    73  // Complex circular arc cosine
    74  //
    75  // DESCRIPTION:
    76  //
    77  // w = arccos z  =  PI/2 - arcsin z.
    78  //
    79  // ACCURACY:
    80  //
    81  //                      Relative error:
    82  // arithmetic   domain     # trials      peak         rms
    83  //    DEC       -10,+10      5200      1.6e-15      2.8e-16
    84  //    IEEE      -10,+10     30000      1.8e-14      2.2e-15
    85  
    86  // Acos returns the inverse cosine of x.
    87  func Acos(x complex128) complex128 {
    88  	w := Asin(x)
    89  	return complex(math.Pi/2-real(w), -imag(w))
    90  }
    91  
    92  // Acosh returns the inverse hyperbolic cosine of x.
    93  func Acosh(x complex128) complex128 {
    94  	w := Acos(x)
    95  	if imag(w) <= 0 {
    96  		return complex(-imag(w), real(w)) // i * w
    97  	}
    98  	return complex(imag(w), -real(w)) // -i * w
    99  }
   100  
   101  // Complex circular arc tangent
   102  //
   103  // DESCRIPTION:
   104  //
   105  // If
   106  //     z = x + iy,
   107  //
   108  // then
   109  //          1       (    2x     )
   110  // Re w  =  - arctan(-----------)  +  k PI
   111  //          2       (     2    2)
   112  //                  (1 - x  - y )
   113  //
   114  //               ( 2         2)
   115  //          1    (x  +  (y+1) )
   116  // Im w  =  - log(------------)
   117  //          4    ( 2         2)
   118  //               (x  +  (y-1) )
   119  //
   120  // Where k is an arbitrary integer.
   121  //
   122  // catan(z) = -i catanh(iz).
   123  //
   124  // ACCURACY:
   125  //
   126  //                      Relative error:
   127  // arithmetic   domain     # trials      peak         rms
   128  //    DEC       -10,+10      5900       1.3e-16     7.8e-18
   129  //    IEEE      -10,+10     30000       2.3e-15     8.5e-17
   130  // The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
   131  // had peak relative error 1.5e-16, rms relative error
   132  // 2.9e-17.  See also clog().
   133  
   134  // Atan returns the inverse tangent of x.
   135  func Atan(x complex128) complex128 {
   136  	x2 := real(x) * real(x)
   137  	a := 1 - x2 - imag(x)*imag(x)
   138  	if a == 0 {
   139  		return NaN()
   140  	}
   141  	t := 0.5 * math.Atan2(2*real(x), a)
   142  	w := reducePi(t)
   143  
   144  	t = imag(x) - 1
   145  	b := x2 + t*t
   146  	if b == 0 {
   147  		return NaN()
   148  	}
   149  	t = imag(x) + 1
   150  	c := (x2 + t*t) / b
   151  	return complex(w, 0.25*math.Log(c))
   152  }
   153  
   154  // Atanh returns the inverse hyperbolic tangent of x.
   155  func Atanh(x complex128) complex128 {
   156  	z := complex(-imag(x), real(x)) // z = i * x
   157  	z = Atan(z)
   158  	return complex(imag(z), -real(z)) // z = -i * z
   159  }
   160  

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