x01.Lab.Numerics: 复幂指数
复数很有用，因为复数运算即向量运算，而向量在游戏、图像处理等场景必不可少。
复数很复杂，想一想复数的复数次方，那就不是一般的晕！
复幂指数离不了欧拉公式:
e^ix = cosx + i sinx; (自然对数底数 e 的 ix 次方 = cosx + i sinx)
欧拉公式的详细解释，可参看网上的维基百科。在此基础上，不妨研究一下复数的复数次方：
public static Complex_R Pow(Complex_R value, Complex_R power) { if (power == Zero) { return One; } if (value == Zero) { return Zero; } double real = value.m_real; double imaginary = value.m_imaginary; double p_real = power.m_real; double p_imaginary = power.m_imaginary; double r = Abs(value); double rad = Math.Atan2(imaginary, real);
double v = (p_real * rad) + (p_imaginary * Math.Log(r)); double u = Math.Pow(r, p_real) * Math.Pow(2.7182818284590451, -p_imaginary * rad); return new Complex_R(u * Math.Cos(v), u * Math.Sin(v)); }
关键的是 u 和 v, 其解释如下：
// r = Abs(x+iy); 即复数的模，向量之长度。 // rad = Math.Atan2(y,x); 即角度，向量之方向。 // log is Math.Log. 求自然对数。 // i 是虚数符号，无为算术优先级所干扰。 // 注1. x+iy = r * (cos(rad)+i sin(rad)) = r * e^i rad // 见维基百科欧拉公式的 在复分析的应用 一节 (x+iy)^(c+id) => e^( log( (x+iy)^(c+id) ) ) // a = e^log(a) => e^( (c+id)*log(x+iy) ) // log(a^b) = b*log(a) => e^((c+id)*log(r* e^i rad)) // 此处用到 注1. => e^((c+id)*(log(r) + i rad)) // 乘法而已 => e^( (c*log(r)-d*rad) + i (c*rad + d*log(r)) ) // v = after i => r^c*e^(-d*rad) * ( cos(v) + i sin(v) ) // u = before * => u * ( cos(v) + i sin(v) )
顺便，把我用 Net Reflector 反编译的 Complex_R 的代码粘贴如下：
Complex_R
namespace x01.Lab.Numerics { using System; using System.Globalization; using System.Runtime; using System.Runtime.InteropServices; [AttributeUsage(AttributeTargets.All, Inherited = false)] internal sealed class __DynamicallyInvokable : Attribute { } [Serializable, StructLayout(LayoutKind.Sequential), __DynamicallyInvokable] public struct Complex_R : IEquatable<Complex_R>, IFormattable { private double m_real; private double m_imaginary; [__DynamicallyInvokable] public static readonly Complex_R Zero; [__DynamicallyInvokable] public static readonly Complex_R One; [__DynamicallyInvokable] public static readonly Complex_R ImaginaryOne; private const double LOG_10_INV = 0.43429448190325; [__DynamicallyInvokable] public double Real { [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] get { return this.m_real; } } [__DynamicallyInvokable] public double Imaginary { [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] get { return this.m_imaginary; } } [__DynamicallyInvokable] public double Magnitude { [__DynamicallyInvokable] get { return Abs(this); } } [__DynamicallyInvokable] public double Phase { [__DynamicallyInvokable] get { return Math.Atan2(this.m_imaginary, this.m_real); } } [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] public Complex_R(double real, double imaginary) { this.m_real = real; this.m_imaginary = imaginary; } [__DynamicallyInvokable] public static Complex_R FromPolarCoordinates(double magnitude, double phase) { return new Complex_R(magnitude * Math.Cos(phase), magnitude * Math.Sin(phase)); } [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] public static Complex_R Negate(Complex_R value) { return -value; } [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] public static Complex_R Add(Complex_R left, Complex_R right) { return (left + right); } [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] public static Complex_R Subtract(Complex_R left, Complex_R right) { return (left - right); } [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] public static Complex_R Multiply(Complex_R left, Complex_R right) { return (left * right); } [__DynamicallyInvokable, TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")] public static Complex_R Divide(Complex_R dividend, Complex_R divisor) { return (dividend / divisor); } [__DynamicallyInvokable] public static Complex_R operator -(Complex_R value) { return new Complex_R(-value.m_real, -value.m_imaginary); } [__DynamicallyInvokable] public static Complex_R operator +(Complex_R left, Complex_R right) { return new Complex_R(left.m_real + right.m_real, left.m_imaginary + right.m_imaginary); } [__DynamicallyInvokable] public static Complex_R operator -(Complex_R left, Complex_R right) { return new Complex_R(left.m_real - right.m_real, left.m_imaginary - right.m_imaginary); } [__DynamicallyInvokable] public static Complex_R operator *(Complex_R left, Complex_R right) { double real = (left.m_real * right.m_real) - (left.m_imaginary * right.m_imaginary); return new Complex_R(real, (left.m_imaginary * right.m_real) + (left.m_real * right.m_imaginary)); } [__DynamicallyInvokable] public static Complex_R operator /(Complex_R left, Complex_R right) { double real = left.m_real; double imaginary = left.m_imaginary; double num3 = right.m_real; double num4 = right.m_imaginary; if (Math.Abs(num4) < Math.Abs(num3)) { double num5 = num4 / num3; return new Complex_R((real + (imaginary * num5)) / (num3 + (num4 * num5)), (imaginary - (real * num5)) / (num3 + (num4 * num5))); } double num6 = num3 / num4; return new Complex_R((imaginary + (real * num6)) / (num4 + (num3 * num6)), (-real + (imaginary * num6)) / (num4 + (num3 * num6))); } [__DynamicallyInvokable] public static double Abs(Complex_R value) { if (double.IsInfinity(value.m_real) || double.IsInfinity(value.m_imaginary)) { return double.PositiveInfinity; } double num = Math.Abs(value.m_real); double num2 = Math.Abs(value.m_imaginary); if (num > num2) { double num3 = num2 / num; return (num * Math.Sqrt(1.0 + (num3 * num3))); } if (num2 == 0.0) { return num; } double num4 = num / num2; return (num2 * Math.Sqrt(1.0 + (num4 * num4))); } [__DynamicallyInvokable] public static Complex_R Conjugate(Complex_R value) { return new Complex_R(value.m_real, -value.m_imaginary); } [__DynamicallyInvokable] public static Complex_R Reciprocal(Complex_R value) { if ((value.m_real == 0.0) && (value.m_imaginary == 0.0)) { return Zero; } return (One / value); } [__DynamicallyInvokable] public static bool operator ==(Complex_R left, Complex_R right) { return ((left.m_real == right.m_real) && (left.m_imaginary == right.m_imaginary)); } [__DynamicallyInvokable] public static bool operator !=(Complex_R left, Complex_R right) { if (left.m_real == right.m_real) { return !(left.m_imaginary == right.m_imaginary); } return true; } [__DynamicallyInvokable] public override bool Equals(object obj) { return ((obj is Complex_R) && (this == ((Complex_R)obj))); } [__DynamicallyInvokable] public bool Equals(Complex_R value) { return (this.m_real.Equals(value.m_real) && this.m_imaginary.Equals(value.m_imaginary)); } [__DynamicallyInvokable] public static implicit operator Complex_R(short value) { return new Complex_R((double)value, 0.0); } [__DynamicallyInvokable] public static implicit operator Complex_R(int value) { return new Complex_R((double)value, 0.0); } [__DynamicallyInvokable] public static implicit operator Complex_R(long value) { return new Complex_R((double)value, 0.0); } [CLSCompliant(false), __DynamicallyInvokable] public static implicit operator Complex_R(ushort value) { return new Complex_R((double)value, 0.0); } [CLSCompliant(false), __DynamicallyInvokable] public static implicit operator Complex_R(uint value) { return new Complex_R((double)value, 0.0); } [CLSCompliant(false), __DynamicallyInvokable] public static implicit operator Complex_R(ulong value) { return new Complex_R((double)value, 0.0); } [CLSCompliant(false), __DynamicallyInvokable] public static implicit operator Complex_R(sbyte value) { return new Complex_R((double)value, 0.0); } [__DynamicallyInvokable] public static implicit operator Complex_R(byte value) { return new Complex_R((double)value, 0.0); } [__DynamicallyInvokable] public static implicit operator Complex_R(float value) { return new Complex_R((double)value, 0.0); } [__DynamicallyInvokable] public static implicit operator Complex_R(double value) { return new Complex_R(value, 0.0); } //[__DynamicallyInvokable] //public static explicit operator Complex(BigInteger value) //{ // return new Complex((double)value, 0.0); //} [__DynamicallyInvokable] public static explicit operator Complex_R(decimal value) { return new Complex_R((double)value, 0.0); } [__DynamicallyInvokable] public override string ToString() { return string.Format(CultureInfo.CurrentCulture, "({0}, {1})", new object[] { this.m_real, this.m_imaginary }); } [__DynamicallyInvokable] public string ToString(string format) { return string.Format(CultureInfo.CurrentCulture, "({0}, {1})", new object[] { this.m_real.ToString(format, CultureInfo.CurrentCulture), this.m_imaginary.ToString(format, CultureInfo.CurrentCulture) }); } [__DynamicallyInvokable] public string ToString(IFormatProvider provider) { return string.Format(provider, "({0}, {1})", new object[] { this.m_real, this.m_imaginary }); } [__DynamicallyInvokable] public string ToString(string format, IFormatProvider provider) { return string.Format(provider, "({0}, {1})", new object[] { this.m_real.ToString(format, provider), this.m_imaginary.ToString(format, provider) }); } [__DynamicallyInvokable] public override int GetHashCode() { int num = 0x5f5e0fd; int num2 = this.m_real.GetHashCode() % num; int hashCode = this.m_imaginary.GetHashCode(); return (num2 ^ hashCode); } [__DynamicallyInvokable] public static Complex_R Sin(Complex_R value) { double real = value.m_real; double imaginary = value.m_imaginary; return new Complex_R(Math.Sin(real) * Math.Cosh(imaginary), Math.Cos(real) * Math.Sinh(imaginary)); } [__DynamicallyInvokable] public static Complex_R Sinh(Complex_R value) { double real = value.m_real; double imaginary = value.m_imaginary; return new Complex_R(Math.Sinh(real) * Math.Cos(imaginary), Math.Cosh(real) * Math.Sin(imaginary)); } [__DynamicallyInvokable] public static Complex_R Asin(Complex_R value) { return (-ImaginaryOne * Log((ImaginaryOne * value) + Sqrt(One - (value * value)))); } [__DynamicallyInvokable] public static Complex_R Cos(Complex_R value) { double real = value.m_real; double imaginary = value.m_imaginary; return new Complex_R(Math.Cos(real) * Math.Cosh(imaginary), -(Math.Sin(real) * Math.Sinh(imaginary))); } [__DynamicallyInvokable] public static Complex_R Cosh(Complex_R value) { double real = value.m_real; double imaginary = value.m_imaginary; return new Complex_R(Math.Cosh(real) * Math.Cos(imaginary), Math.Sinh(real) * Math.Sin(imaginary)); } [__DynamicallyInvokable] public static Complex_R Acos(Complex_R value) { return (-ImaginaryOne * Log(value + (ImaginaryOne * Sqrt(One - (value * value))))); } [__DynamicallyInvokable] public static Complex_R Tan(Complex_R value) { return (Sin(value) / Cos(value)); } [__DynamicallyInvokable] public static Complex_R Tanh(Complex_R value) { return (Sinh(value) / Cosh(value)); } [__DynamicallyInvokable] public static Complex_R Atan(Complex_R value) { Complex_R complex = new Complex_R(2.0, 0.0); return ((ImaginaryOne / complex) * (Log(One - (ImaginaryOne * value)) - Log(One + (ImaginaryOne * value)))); } [__DynamicallyInvokable] public static Complex_R Log(Complex_R value) { return new Complex_R(Math.Log(Abs(value)), Math.Atan2(value.m_imaginary, value.m_real)); } [__DynamicallyInvokable] public static Complex_R Log(Complex_R value, double baseValue) { return (Log(value) / Log(baseValue)); } [__DynamicallyInvokable] public static Complex_R Log10(Complex_R value) { return Scale(Log(value), 0.43429448190325); } [__DynamicallyInvokable] public static Complex_R Exp(Complex_R value) { double num = Math.Exp(value.m_real); double real = num * Math.Cos(value.m_imaginary); return new Complex_R(real, num * Math.Sin(value.m_imaginary)); } [__DynamicallyInvokable] public static Complex_R Sqrt(Complex_R value) { return FromPolarCoordinates(Math.Sqrt(value.Magnitude), value.Phase / 2.0); } [__DynamicallyInvokable] public static Complex_R Pow(Complex_R value, Complex_R power) { if (power == Zero) { return One; } if (value == Zero) { return Zero; } double real = value.m_real; double imaginary = value.m_imaginary; double p_real = power.m_real; double p_imaginary = power.m_imaginary; double r = Abs(value); double rad = Math.Atan2(imaginary, real); double v = (p_real * rad) + (p_imaginary * Math.Log(r)); double u = Math.Pow(r, p_real) * Math.Pow(2.7182818284590451, -p_imaginary * rad); return new Complex_R(u * Math.Cos(v), u * Math.Sin(v)); } [__DynamicallyInvokable] public static Complex_R Pow(Complex_R value, double power) { return Pow(value, new Complex_R(power, 0.0)); } private static Complex_R Scale(Complex_R value, double factor) { double real = factor * value.m_real; return new Complex_R(real, factor * value.m_imaginary); } static Complex_R() { Zero = new Complex_R(0.0, 0.0); One = new Complex_R(1.0, 0.0); ImaginaryOne = new Complex_R(0.0, 1.0); } internal static void Test() { var c1 = new Complex_R(3.0, 4.0); var c2 = new Complex_R(2.0, 0.0); Console.WriteLine(Complex_R.Pow(c1, c2)); Console.WriteLine(c1 * c1 ); } } }
有兴趣的不妨在控制台程序中运行一下，我相信会获益良多的。