Logo_Math
Suitcase of Dreams
Math, database, programming, graphics, web design
Logo_Integral
Complex functions Tutorial
Complex analysis studies the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of math of real numbers may break down when applied to complex numbers.
Free Lessons

Lesson 1 Complex numbers
In this lesson: Forms and representations of the complex numbers; Modulus and arguments; Principal value of the argument.

Lesson 2 Trigonometric and algebraic form conversion
In this lesson: Complex numbers forms conversion; Examples of the conversion.

Lesson 3 The algebra of complex numbers
In this lesson: Arithmetic operations with complex numbers; Properties of the complex numbers; Geometric interpretation of addition & subtraction.

Lesson 4 Geometric interpretation of multiplication
In this lesson: The modulus and argument of the product; Multiplication of complex numbers as stretching - squeezing and rotation; Multiplying a complex number by imaginary unit i and by powers of i.

Lesson 5 Division of the complex numbers
In this lesson: Definition and notation of conjugates and reciprocals; Division as multiplication and reciprocation.

Lesson 6 Powers and roots of complex numbers
In this lesson: De Moire's theorem; Powers of complex numbers; n-th root of complex numbers.

Lesson 7 Complex Exponential Function and Complex Logarithm Function
In this lesson: Definition and notation; Complex logarithm function is a multi-valued function; Principal branch of the logarithm.

Lesson 8 Complex Trigonometric Functions and Complex Inverse Trigonometric Functions
In this lesson: A difference between the real and complex trigonometric functions; Relationship to exponential function; Identities; Derivatives and Indefinite integrals of inverse trigonometric functions.

Lesson 9 Complex Hyperbolic Functions and Inverse Hyperbolic Functions
In this lesson: The notations; Definitions; Derivatives and Indefinite integrals of inverse hyperbolic functions.

Lesson 10 Complex Power Function
In this lesson: Raising a complex number to a complex power; Derivatives and Indefinite integral of complex power function.

Lesson 11 Complex Rational Functions
In this lesson: Definition of the rational function; Möbius transformations; Fractional-linear function; Zhukovskii function.

Complex Power Function

Complex Analysis. FreeTutorial

1) Computation of complex power function
     1.1 The complex power function is a multi-valued function
     1.2 Computation of the roots of the complex value

     1.3 Computation of the complex z and real a
     1.4 The power of a real number to a non-integer power    

     1.5 Raising a complex number to a complex power

2) The rules for combining quantities containing powers

3) Derivatives of complex power function

4) Indefinite integral of complex power function

1) Computation of the complex power function

1.1 The complex power function is a multi-valued function

Computation of complex power function involves using the complex Exponential and Logarithm functions.
For any z and every a

complex power function formula

where zC.

Because Ln(z) is a multi-valued function, the function is a multi-valued too. The principal branch of the function is obtained by replacing Ln(z) with the principal branch of the logarithm.

The power a may be an integer, real number, or complex number.

A number other than "0" taken to the power "0" is defined to be 1, which follows from the
A number to the first power is, by definition, equal to itself:
z
1
= z.

Top

1.2 Computation of the roots of the complex value

If  a = , nN, then

root of the complex value formula

Top

1.3 Computation of the complex z and real a

For complex z and real a,

complex power function formula
where arg(z) is the complex argument.

Top

1.4 The power of a real number to a non-integer power

The power of a real number to a non-integer power is not necessarily itself a real number.
For example, is real only for x0.

Top

1.5 Raising a complex number to a complex power

A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies

raising a complex number to a complex power formula

where arg(z) is the complex argument.

Written explicitly in terms of real and imaginary parts,

raising a complex number to a complex power formula continue

Example of complex exponentiation:

complex exponentiation example

The result of raising a complex number to a complex power may be a real number.

Example
Let us find a principle value of= 0.20787957635

In fact, there is a family of values k such that is real:
[Cos(k ln k) + iSin(k ln k)]

This will be real when Sin(k ln k) = 0, i.e., for

k ln k = n

Top

The rules for combining quantities containing powers

Some of the rules for combining quantities containing powers carry over from the real case. If c and d are complex numbers, and , then

; (2.1)
; (2.2)
; (2.3)
, where n is real integer. (2.4)

The identity (2.4) does not hold if n is replaced with an arbitrary complex value.

Example:

example formula, k is an integer.

, k is an integer.

Since these sets of solutions are not equal, identity (2.4) does not always hold.

The principal values of and are the same.
The principal value of = 0.0432139
The principal value of = 0.0432139

Top

3) Derivatives of power function

3.1 derivative formula

For the proof, we use the definition where zC. Then we have:

derivative formula 1

3.2 derivative formula 2

For the proof, we use the definition where zC. Then we have:

derivative formula 3

Top

4) Indefinite integral of power function

The indefinite integral:

indefinite integral formula

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
Set theory paradoxes
We will add more
Copyright © 2006 Suitcase of Dreams All rights reserved.