Baselines Designs – Music Production and Creation



Complex Variables – Formulae, Tips and Tricks

Steve Keith –

Abstract

This is an ongoing project – last update was Mar 26, 2010. Check back for new additions weekly.

Formulae

Quadratic Equation for Complex Numbers - for the quadratic equation [pic]

Deriving z by completing the square:

|[pic] |

|[pic] |

|[pic] |

|What term (X) is needed to complete the square? |

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|[pic] so [pic] |

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|[pic] |

|[pic] |

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| [pic] |

[pic]

Or, when factoring out the square root of -1 to emphasize the complex nature:

[pic]

The variables a, b and c can all be complex numbers and this will still work.

Binomial Theorem

For n>0

[pic]

For any integer

[pic]

Some properties of Complex Numbers

[pic]

[pic] = modulus

[pic] = [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Triangle Inequality Formulae

[pic]

[pic]

[pic]

Polar Magnitude and Angle

[pic]

[pic]

The principal argument of the multi-valued argument is between –[pic] and +[pic]

For a circle:

[pic]

[pic]

Polar Form of Complex Number

[pic]

DeMoivre’s Theorem (extended to show complex number to a power)

[pic]

Complex Number to a Fractional Power

[pic]

Riemann Number Sphere

Any number in the complex plane can be represented by the tangent point on the unit sphere of a line drawn from the top of the unit sphere to the representation of the complex number in the xy plane that the sphere sits on.

x and y in terms of z

[pic]

[pic]

Definition:

[pic]

Using the above definitions, you can map a function from the z plane to the w plane.

The Complex Derivative (definition)

[pic]

Derivation of the Cauchy Riemann Equations

In order for a complex function to exist at a point, the Cauchy Riemann Equations must be satisfied. Defining f(z) as a function of a real (u) and an imaginary (v) part, and having both function u and v be defined in terms of x and y:

[pic]

[pic]

The derivative at any point that exists must be the same no matter which way you approach the point. We will approach the point [pic] holding y constant so [pic] .

[pic]

Grouping real and imaginary parts together:

[pic]

Recognizing the definition of partial derivatives and passing to the limit:

[pic]

Going through the above derivation, but this time approaching in the y direction, we find:

[pic]

Since [pic] must be the same from whichever way you approach it if the derivative exists at that point, these two results can be equated. (the real part of one must equal the real part of the other and the imaginary part of one must equal the imaginary part of the other.) This leads to the Cauchy Riemann equations:

[pic]

[pic]

If these equations do not hold, then the derivative at the point [pic] does not exist.

If these equations hold AND all the partial derivatives of u and v are continuous in the neighborhood of [pic] then [pic] exists.

All of the normal operations for differentiating real functions work with complex functions.

If a complex function is made up of other differentiable complex functions using addition, subtraction, multiplication, division or powers then the full complex function is differentiable.

L’Hopital’s Rule

For 2 functions g(z) and h(z) that are differentiable at [pic] and

If g([pic])and h([pic]) are both 0 and

If h’([pic]) is NOT equal to 0 then

[pic]

Extension to this rule: if g(z), h(z), and their first n derivatives vanish at [pic], then

[pic]

Analyticity

f(z) is analytic at a point if f’(z) exists at that point and every point in some neighborhood of the point. The sum, difference, product and quotient of two or more analytic functions is an analytic function as long as the denominator of the quotient is not 0.

A function that is analytic throughout the whole complex plane is called an entire function.

Derivation of Cauchy Riemann equations in terms of r and [pic]

[pic]

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[pic] [pic]

[pic] [pic]

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Polar Form of CR Equations

[pic]

[pic] , [pic]

[pic] , [pic]

Chain Rule Usage

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Substitute the above into the CR equality equations

[pic] , [pic]

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Multiply the first by [pic] and multiply the second by [pic]

[pic]

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Add these two equations together

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Simplifying

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And taking advantage of [pic]

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A similar derivation leads to the companion equation:

[pic]

Using

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Multiply both sides by [pic]

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From Above:

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So

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Similarly

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So

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Factoring

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Or using the other equation

[pic]

So, using either of the two above equations (r not equal to zero), you can find the derivative at a point if it exists using polar form.

LaPlace’s Equation

[pic]

Derivation of LaPlace’s Equation – start with CR equations

[pic], [pic]

So

[pic] , [pic]

Adding

[pic]

This holds for v(x,y) as well combining both u and v parts gets you to LaPlace’s Equation.

Harmonic Functions

Any function satisfying LaPlace’s equation is said to be harmonic.

Wherever a function is analytic, its real and imaginary parts are harmonic.

The real and imaginary parts of harmonic functions are call cunjugates of one another.

There are lines of constant value that make up curves for the real part of a complex harmonic function, these are called equi-potentials or iso-therms or something similar depending on the application. There are also constant value lines when the imaginary part is considered. The imaginary family of constant value lines are called streamlines. Streamlines are orthogonal to equipotentials (perpendicular). If a complex harmonic function defines a vector field, such as an electric field, then the direction of the vector at every point is tangential to the streamline curves.

Definition: Total Differential (variables x and y)

[pic]

On equipotentials and streamlines, u and v are constants, so du or dv will be 0.

Transcendental Complex Functions

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Derivatives of complex transcendental functions follow the same rules as real transcendental functions.

[pic]

The derivative of the real part of a complex function (of a real variable) is the real part of the derivative of the function.

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Exponentials

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Inverse Trig and Hyperbolic Functions

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