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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 sine // // DESCRIPTION: // // If // z = x + iy, // // then // // w = sin x cosh y + i cos x sinh y. // // csin(z) = -i csinh(iz). // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // DEC -10,+10 8400 5.3e-17 1.3e-17 // IEEE -10,+10 30000 3.8e-16 1.0e-16 // Also tested by csin(casin(z)) = z. // Sin returns the sine of x. func Sin(x complex128) complex128 { switch re, im := real(x), imag(x); { case im == 0 && (math.IsInf(re, 0) || math.IsNaN(re)): return complex(math.NaN(), im) case math.IsInf(im, 0): switch { case re == 0: return x case math.IsInf(re, 0) || math.IsNaN(re): return complex(math.NaN(), im) } case re == 0 && math.IsNaN(im): return x } s, c := math.Sincos(real(x)) sh, ch := sinhcosh(imag(x)) return complex(s*ch, c*sh) } // Complex hyperbolic sine // // DESCRIPTION: // // csinh z = (cexp(z) - cexp(-z))/2 // = sinh x * cos y + i cosh x * sin y . // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // IEEE -10,+10 30000 3.1e-16 8.2e-17 // Sinh returns the hyperbolic sine of x. func Sinh(x complex128) complex128 { switch re, im := real(x), imag(x); { case re == 0 && (math.IsInf(im, 0) || math.IsNaN(im)): return complex(re, math.NaN()) case math.IsInf(re, 0): switch { case im == 0: return complex(re, im) case math.IsInf(im, 0) || math.IsNaN(im): return complex(re, math.NaN()) } case im == 0 && math.IsNaN(re): return complex(math.NaN(), im) } s, c := math.Sincos(imag(x)) sh, ch := sinhcosh(real(x)) return complex(c*sh, s*ch) } // Complex circular cosine // // DESCRIPTION: // // If // z = x + iy, // // then // // w = cos x cosh y - i sin x sinh y. // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // DEC -10,+10 8400 4.5e-17 1.3e-17 // IEEE -10,+10 30000 3.8e-16 1.0e-16 // Cos returns the cosine of x. func Cos(x complex128) complex128 { switch re, im := real(x), imag(x); { case im == 0 && (math.IsInf(re, 0) || math.IsNaN(re)): return complex(math.NaN(), -im*math.Copysign(0, re)) case math.IsInf(im, 0): switch { case re == 0: return complex(math.Inf(1), -re*math.Copysign(0, im)) case math.IsInf(re, 0) || math.IsNaN(re): return complex(math.Inf(1), math.NaN()) } case re == 0 && math.IsNaN(im): return complex(math.NaN(), 0) } s, c := math.Sincos(real(x)) sh, ch := sinhcosh(imag(x)) return complex(c*ch, -s*sh) } // Complex hyperbolic cosine // // DESCRIPTION: // // ccosh(z) = cosh x cos y + i sinh x sin y . // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // IEEE -10,+10 30000 2.9e-16 8.1e-17 // Cosh returns the hyperbolic cosine of x. func Cosh(x complex128) complex128 { switch re, im := real(x), imag(x); { case re == 0 && (math.IsInf(im, 0) || math.IsNaN(im)): return complex(math.NaN(), re*math.Copysign(0, im)) case math.IsInf(re, 0): switch { case im == 0: return complex(math.Inf(1), im*math.Copysign(0, re)) case math.IsInf(im, 0) || math.IsNaN(im): return complex(math.Inf(1), math.NaN()) } case im == 0 && math.IsNaN(re): return complex(math.NaN(), im) } s, c := math.Sincos(imag(x)) sh, ch := sinhcosh(real(x)) return complex(c*ch, s*sh) } // calculate sinh and cosh func sinhcosh(x float64) (sh, ch float64) { if math.Abs(x) <= 0.5 { return math.Sinh(x), math.Cosh(x) } e := math.Exp(x) ei := 0.5 / e e *= 0.5 return e - ei, e + ei }