Maths - Functions Of A Complex Variable - Martin Baker

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Maths - Functions Of A Complex Variable

On this page we discuss how to extend the scalar functions (that we discussed on this page) to the situation where the domain and/or codomain are complex numbers, we also discuss functions like normalise which are specific to complex numbers.

• Conjugate
• Powers of complex numbers
• Exponential Function
• log Function
• Inverse Function
• Reflection in circle
• Norm (normalise)

We can categorise these functions according to the the following properties:

• Conformal - preserves angles
• Anticonformal - reverses angles
• Non-conformal - does not conserve angles

preservation of angles	preservation of distances	Analytic
Conjugate	Anticonformal
Powers of complex numbers
Exponential Function	Riemann surface
Inverse Function	Conformal	yes, except at 0 and &in
Reflection in circle	Anticonformal
Norm (normalise)
Möbius transformation

Evaluating Complex Functions

We can evaluate the value of many complex functions by using infinite series in the same way that we can for scalar functions. We just use the same series that we would use for scalar functions but plug in complex values instead of real values. So if a series exists for real values we can evaluate the complex function. For example:

ln(1+z)	z - z 2/2! + z 3/3! ... +(-1)r+1zr/(r)!	-1 < z <= 1
exp(z)	1 + z 1/1! + z 2/2! + z 3/3! ... + zr/(r)!	all values of z

Alternatively we can derive the real and imaginary parts in terms of the real functions:

function of complex variable	real and imaginary terms
exp(x+i y)	exp(x)*(cos(y)+i sin(y))
ln(x+i y)	0.5*ln(x²+y²) + i atan(x/y)
sin(x+i y)	cosh(-y) sin(x) - i sinh(-y) cos(x)
cos(x+i y)	cosh(-y) cos(x) + i sinh(-y) sin(x)
tan(x+i y)	(tan(x) -i tanh(-y)) / (1 +i tanh(-y) tan(x))
sinh(x+i y)	sinh(x) cos(y) + i cosh(x) sin(y)
cosh(x+i y)	cosh(x) cos(y) + i sinh(x) sin(y)
tanh(x+i y)	(tanh(x) + i tan(y)) / (1 +i tanh(x) tan(y))

We can convert between the circular and hyperbolic functions as follows:

• sin(z) = -i sinh(iz)
• cos(z) = -cosh(iz)
• tan(z) = -i tanh(iz)

See also the trig identities on this page.

Conjugate

When we multiply a complex number by its conjugate we get a real number, in other words the imaginary part cancels out.

Inverse Function

To calculate the inverse value (1/z) we multiply the top and bottom by the conjugate which makes the denominator a real number.

z plane	w plane
	--> w=1/z

how this plot was produced.

Let the components of the input and output planes be:

z = x + i y and w = u + i v

In this case w = 1/z

so:

w = 1/(x + i y)

As usual, we evaluate the inverse by multiplying top and bottom by the conjugate:

w = (x - i y)/(x + i y)(x - i y)

w = (x - i y)/(x² + y²)

so the u and v components are:

u = x /(x²+y²) v = -y /(x²+y²)

This function has some very interesting and useful properties:

• Straight lines through the origin map to themselves.
• Straight lines not through the origin map to circles.
• Circles through the origin map to straight lines.
• Circles not through the origin map to circles.
• Applying the inverse function twice restores the original.
• The inverse function can be decomposed into a conjugate and a reflection in a circle.

So this function can be used to map between circles and lines. We can also extend the concept to 3 dimensions which allows us to map the surface of a sphere to the plane (see stereographic projections).

Möbius Transform

This can be generalised to the Möbius Transform (described here):

Where:

• z = input (complex values)
• w = output (complex values)
• a,b,c,d = parameters (complex values)

The inverse function is a special case where a=d=0 and b=c=1 we get:

w=1/z

If we now allow d to be non-zero we can see that this will just shift the line in the z-plane:

w=1/(z+d)

If we now allow c to scale we can see that this will scale the position on the line in the z-plane:

w=1/(cz+d)

If we now allow a and b to vary things get a bit more complicated but we can see that we can move the circle around the w plane.

Exponential Function

There are two methods to calculate the exponential function:

• polar form and Euler's equation as discussed on this page.
• infinite series as discussed on this page.

z plane	w plane
	--> w=ez

how this plot was produced.

Let the components of the input and output planes be:

z = x + i y and w = u + i v

In this case w = ez

so:

w = e(x + i y)

by the rules of exponents:

w = exei y

applying Euler's equation we get:

w = ex(cos(y) + i sin(y))

so the u and v components are:

u = excos(y) v = exsin(y)

Log Function

The natural log, that is the log to the base e, is the inverse of the exponetial function. So if,

w = ln(z)

then taking the exponent of each side gives,

ew = z

so we have

x = eucos(v) y = eusin(v)

which gives:

u = ln(√(x² + y²)) v = atan(x/y)

Powers of complex numbers

Here is the plot for (x + i y)² the more general expression for (x + i y)n is given on this page.

z plane	w plane
	--> w=z²

how this plot was produced.

Pure real values always square to a positive value and pure imaginary values always square to a negative value. However real and imaginary parts together cover the whole plane.

Let the components of the input and output planes be:

z = x + i y and w = u + i v

lets take the example of the square function w = z²

so:

w = (x + i y)²

multiplying out gives:

w = x² - y² + i 2 x y

so the u and v components are:

u = x² - y² v = 2 x y

Norm

This is the distance (r) of a + i b from the origin.

It is written as:

r = | a + i b |

by Pythagoras:

r = | a + i b | = math.sqrt(a*a + b*b)

Check that:

|a + i b|*|c + i d| = |a*c - b*d + i (a*d + b*c)|

Möbius transformation

The mapping between the surface of the sphere and the plane can be represented by the Möbius transformation of the form:

M(z)=	az + b
	cz + d

Where:

• z = complex variable
• a,b,c & d = complex constants

This is useful in the stereographic projection explained on this page.

Further Study

I think its interesting to compare these graphs for complex numbers to the equivalent graphs for double numbers and dual numbers.

EuclideanSpace

home

Prerequisites

On this page we discussed how to represent and evaluate functions where the domain and codomain are complex numbers.

Transforms

A transform maps every point in a vector space to a possibly different point.

When transforming a computer model we transform all the vertices.

To model this using mathematics we can use matrices, quaternions or other algebras which can represent multidimensional linear equations.

This page explains this.

Direction vs. Rotation

A 3D vector can be used to define a direction. If a vector is used to define direction in this way then the length of the vector is not relevant, therefore we can use a unit length vector.

Note that direction is different from rotation (or orientation) in that direction just tells us which direction an object is in but rotation also includes revolving around the vector. So direction would be enough to locate a given star in the night sky but rotation will orient it so the constalations are seen a certain way round.

This page explains this.

For the rotation required to look in a given direction see this page.

Trigonometry Functions

The trigometric functions, such as sin, cos and tan, represent how the ratio of these lengths depends on the angle θ and we can define the following functions for each of the ratios:

sine function	sin(θ) = b / a
cosine function	cos(θ) = c / a
tangent function	tan(θ) = b / c
cosecant	cosec(θ) = a / b
secant	sec(θ) = a / c
cotangent	cot(θ) = c / b

This page explains this.

Standards

There are a lot of choices we need to make in mathematics, for example,

• Left or right handed coordinate systems.
• Vector shown as row or column.
• Matrix order.
• Direction of x,y and z coordinates.
• Euler angle order
• Direction of positive angles
• Choice of basis for bivectors
• Etc. etc.

A lot of these choices are arbitrary as long as we are consistent about it, different authors tend to make different choices and this leads to a lot of confusion. Where standards exist I have tried to follow them (for example x3d and MathML) otherwise I have at least tried to be consistent across the site. I have documented the choices I have made on this page.

Spinor

A spinor represents a transform where a rotation of 360° inverts the object. To restore the object to its original orientation we need to rotate by 720°.

This page explains this.

metadata block
see also:	Conformal Space Stereographic Projection - Riemann Sphere - representing points on a sphere by using complex numbers
Correspondence about this page	Open Forum
Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.	Visual Complex Analysis - If you already know the basics of complex numbers but want to get an in depth understanding using an geometric and intuitive approach then this is a very good book. The book explains how to represent complex transformations such as the Möbius transformations. It also shows how complex functions can be differentiated and integrated.
Terminology and Notation Specific to this page here:

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