Mathematics Database Programming Web Design Price List     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

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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

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 where z C.

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.

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1.2 Computation of the roots of the complex value

If  a = , n N, then Top

1.3 Computation of the complex z and real a

For complex z and real a, where arg(z) is the complex argument.

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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 x 0.

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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 where arg(z) is the complex argument.

Written explicitly in terms of real and imaginary parts, Example of complex exponentiation: 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:  , 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

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3) Derivatives of power function

3.1 For the proof, we use the definition where z C. Then we have: 3.2 For the proof, we use the definition where z C. Then we have: Top

4) Indefinite integral of power function

The indefinite integral: Previous theme Content Next theme

by Tetyana Butler Top  Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Paradox of even (odd) and natural numbers Paradox of Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise Contact us