NMath User's Guide

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3.5 Functions of Complex Numbers (.NET, C#, CSharp, VB, Visual Basic, F#)

NMath provides a variety of functions that take complex numbers as arguments.

Conjugate, Norm, and Argument

NMath provides static functions on FloatComplex and DoubleComplex for common complex number functions:

● The static Conj() function returns the conjugate of a complex number. The conjugate of a complex number a + bi is defined as ­a - b­i.

● The static Norm() method returns the norm (or modulus) of a complex number, defined as the square root of the sum of the squares of the real and imaginary parts.

● The static Arg() method returns the argument of a complex number, defined as the directed phase angle in polar coordinates.

For instance:

Code Example – C# complex numbers

var c = new FloatComplex( -8.2, 3.4 );

FloatComplex conj = FloatComplex.Conj( c );

float norm = FloatComplex.Norm( c );

float arg = FloatComplex.Arg( c );

Code Example – VB complex numbers

Dim C As New FloatComplex(-8.2, 3.4)

Dim Conj = FloatComplex.Conj(C)

Dim Norm = FloatComplex.Norm(C)

Dim Arg = FloatComplex.Arg(C)

Trigonometric Functions

NMath extends standard trigonometric functions Sin(), Cos(), Sinh(), Cosh(), Tan(), and Tanh() to take complex number arguments. Class NMathFunctions provides these functions as static methods; all take a single complex number as an argument and return a complex number as a result:

Code Example – C# complex numbers

var c = new DoubleComplex( 1.0, -3.9 );

DoubleComplex sin = NMathFunctions.Sin( c );

DoubleComplex cos = NMathFunctions.Cos( c );

Code Example – VB complex numbers

Dim C As New DoubleComplex(1.0, -3.9)

Dim Sin = NMathFunctions.Sin(C)

Dim Cos = NMathFunctions.Cos(C)

Transcendental Functions

NMath extends standard transcendental functions Exp() and Log() to take complex arguments. Class NMathFunctions provides these functions as static methods. For example:

Code Example – C# complex numbers

var c = new FloatComplex( -8.11, 3.04 );

FloatComplex exp = NMathFunctions.Exp( c );

FloatComplex log = NMathFunctions.Log( c );

Code Example – VB complex numbers

Dim C As New FloatComplex(-8.11, 3.04)

Dim Exp = NMathFunctions.Exp(C)

Dim Log = NMathFunctions.Log(C)

Class NMathFunctions also provides several static overloads of the exponential function Pow(). Versions exist to:

● raise a complex number to an integer exponent

● raise a complex number to a real exponent

● raise a complex number to a complex exponent

● raise a real value to a complex exponent

All return a complex number. For instance:

Code Example – C# complex numbers

var c1 = new DoubleComplex( 12.932, -4.0 );

DoubleComplex c2 = NMathFunctions.Pow( c1, 3 );

DoubleComplex c3 = NMathFunctions.Pow( c1, 1.12 );

DoubleComplex c4 = NMathFunctions.Pow( c1, c3 );

DoubleComplex c5 = NMathFunctions.Pow( 5.2, c1 );

Code Example – VB complex numbers

Dim C1 As New DoubleComplex(12.932, -4.0)

Dim C2 = NMathFunctions.Pow(C1, 3)

Dim C3 = NMathFunctions.Pow(C1, 1.12)

Dim C4 = NMathFunctions.Pow(C1, C3)

Dim C5 = NMathFunctions.Pow(5.2, C1)

Absolute Value and Square Root

The static Abs() function on class NMathFunctions returns the absolute value of a complex number, which is simply equal to the norm:

Code Example – C# complex numbers

var c = new DoubleComplex( 7.99, 0.3 );

double abs = NMathFunctions.Abs( c );

Code Example – VB complex numbers

Dim C As New DoubleComplex(7.99, 0.3)

Dim Abs = NMathFunctions.Abs(C)

NMath also extends the standard Sqrt() function to take a complex argument, again as a static method on class NMathFunctions. For example:

Code Example – C# complex numbers

var c = new FloatComplex( -8.11, 3.04 );

FloatComplex sqrt = NMathFunctions.Sqrt( c );

Code Example – VB complex numbers

Dim C As New FloatComplex(-8.11, 3.04)

Dim Sqrt = NMathFunctions.Sqrt(C)

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