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Complex Numbers

Complex numbers arise in algebra in the solution of quadratic equations. The equation x² = 1 does not have any real solutions. However, if we define a new number, i as the square root of -1, then we have two solutions: i and -i. This, in turn, gives rise to an entirely new class of numbers of the form a + ib with a and b real, and i defined as above. These are the complex numbers.

The Extreme Optimization Mathematics Library for .NET provides value types for single and double precision complex numbers. The DoubleComplex value type represents double-precision complex numbers. Value types are implemented using structures.

Constructing complex numbers

The DoubleComplex structure has three constructors.

The first constructor takes two arguments: the real and imaginary parts of the complex number.

The second constructor takes one real argument. It constructs a complex number whose imaginary part is zero.

The third constructor takes three arguments: two real numbers and a boolean parameter named polar. If polar is true, the first two arguments are treated as the modulus and argument of the complex number. If polar is false, the first two arguments are treated as the real and imaginary part of the complex number.

Finally, the static RootOfUnity method constructs a complex number that is one of the solutions to the equation xⁿ = 1. The solutions to this equation are n complex numbers spaced equally on the unit circle. The first parameter of the method is the degree, n, of the root. The second parameter is the index of the root in the series, counting counter-clockwise on the unit circle. An index of 0 corresponds to the root x = 1.

DoubleComplex constants

The complex structure provides several constants for commonly used and special complex numbers. These are listed in the following table:

Table 1. Complex number constants.

Two values in the above list deserve special attention. The Infinity field represents complex infinity. It is the result of dividing any non-zero complex number by zero. DoubleComplex infinity does not have a sign. Because the complex numbers do not have a natural ordering, is does not make sense to speak of positive or negative numbers. Directed infinities, where the modulus of the complex number is infinite, but its argument is not, are not supported. The static IsInfinity method verifies whether a complex number equals DoubleComplex.Infinity .

The NaN field represents complex Not-a-Number. It is a special value that is the result of dividing zero by zero.The static IsNaN method verifies whether a complex number equals DoubleComplex.NaN. You can not use the equality operator for this purpose, as it always returns false.

Working with complex numbers

Working with complex numbers is easy. The Re property gets the real component of the complex number, while Im gets the imaginary part. The Modulus property returns the square root of the sum of the squares of the real and imaginary components. The Argument property returns the angle between the positive real axis and a line from the origin to the complex number, measured counter-clockwise.

The modulus is computed every time the Modulus property is called. The same is true for the Argument property. If you use these values multiple times, you should consider caching them in a variable.

Details of complex arithmetic

When performing binary operations, if one of the operands is a DoubleComplex, then the other operand is required to be an integral type or a floating-point type (Double or Single) or a complex type (DoubleComplex). Prior to performing the operation, if the other operand is not a DoubleComplex, it is converted to DoubleComplex, and the operation is performed using at least DoubleComplex range and precision. If the operation produces a numerical result, the type of the result is DoubleComplex. Exceptions to this are methods that return a real property of a number: Argument, Modulus, ModulusSquared and Abs. In these cases, the return type is Double.

The Extreme Optimization Mathematics Library for .NET extends the IEEE-754 standard definition of NaN's for real numbers to complex numbers as follows.

The floating-point operators, including the assignment operators, do not throw exceptions. Instead, in exceptional situations, the result of a floating-point operation is zero, Infinity, or NaN, as described below:

• If the result of a complex floating-point operation is too small for the destination format, the result of the operation is zero.
• If the magnitude of the complex result of a floating-point operation is too large for the destination format, the result of the operation is DoubleComplex.Infinity.
• If a complex floating-point operation on complex numbers is invalid, the result of the operation is DoubleComplex.NaN.
• If one or both operands of a complex floating-point operation are NaN (real or complex), the result of the operation is DoubleComplex.NaN.

Arithmetic operations

The Extreme Optimization Mathematics Library for .NET provides methods for all basic arithmetic operators on complex numbers. Most contain special versions for cases where one of the operands is real. Overloaded versions of the arithmetic operators are provided for languages that support them. For languages that don't support operator overloading, equivalent static (Shared in Visual Basic) methods are supplied.

For example, here is some C# code:

[C#]
DoubleComplex a = new DoubleComplex(1, 2);
DoubleComplex b = new DoubleComplex(-3, 4);
DoubleComplex c = 2 - 1 / (a + b);

and here is the equivalent Visual Basic.NET code:

[Visual Basic]
Dim a As DoubleComplex = New DoubleComplex(1, 2)
Dim b As DoubleComplex = New DoubleComplex(-3, 4)
Dim c As DoubleComplex = _
DoubleComplex.Subtract(2, DoubleComplex.Divide(1, DoubleComplex.Add(a, b)))

Specific to complex numbers is the static Conjugate method. This method returns the conjugate of a complex number, leaving the original unchanged.

Table 2. Complex number operators and their static (Shared) method equivalents.

There is one other static method that does not have an operator equivalent. The ConjugateMultiply method returns the product of the conjugate of the first argument and the second argument.

Because the complex numbers don't have a natural ordering, only equality and inequality operators are available. No other comparison operators are available.

Functions of complex numbers

The DoubleComplex type defines static methods for the most common mathematical functions of complex numbers, including: logarithmic, exponential, trigonometric and hyperbolic functions.

Some functions of real numbers have a limited domain when the result is restricted to be real, but are defined for all real numbers if the result can be complex. Examples of this are the inverse sine and cosine, which are real only for arguments between -1 and +1. The DoubleComplex type defines methods that extend these functions to return a complex result.

These functions are often discontinuous along the part of the real axis where they are complex-valued. Which of the two limit values is chosen is arbitrary. However, the most consistent choice is the one that preserves any symmetry or anti-symmetry about the origin. For example, the function Arcsin(x) satisfies Arcsin(-x) = -Arcsin(x) when x is within [-1, 1]. The Arcsin method with real argument also satisfies this identity.

Most elementary functions have been extended to cover the entire complex plane. Once again, many of them are multi-valued, and a suitable choice has to be made regarding which of the two possible limit values is returned. The choice is made easier by the fact that the discontinuities lie along the real or imaginary axis. This makes it possible to distinguish between the 'limit from above' and the 'limit from below' by using the value of negative zero for the limit from below. The situation is somewhat simplified by the fact that any analytic function satisfies the identity f(conj(z) = conj(f(z)).

The tables below summarize these methods, and their meaning. Special functions with complex argument may be available from the Extreme.Mathematics.SpecialFunctions namespace.

Table 3. Logarithmic and exponential functions of complex numbers.

Table 4. Trigonometric functions of complex numbers.

Table 5. Hyperbolic functions of complex numbers.

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