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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 Logarithm Function

Complex Analysis. FreeTutorial

The logarithm function Ln(z) is an inverse of the exponential function. Because the complex exponential function is a periodic function

  Proof
an equation  f(z)= has infinitely many solutions in a case of complex variable, and the complex logarithm function Ln(z) is a multi-valued function.

Ln(z) = ln(|z|) + i[arg(z) +], k = 0, ±1, ±2, ...

If k = 0 we have a principal logarithm ln(z) or principal branch of the logarithm:

ln(z) = ln(|z|) + i[arg(z)]

Example 1

Find the values of

1) ln(-5)

2) Ln(-5),

where ln(-5) is a principal logarithm.

1) ln(-5) = ln(|-5|) + i[arg(-5)] = ln(5) +

2) Ln(-5) = ln(|-5|) + i[arg (-5)] + =

= ln(5) + (2k + 1), where k – any integer.

The logarithm function Ln(z) has a singularity at z = 0. If the non-zero complex number z is expressed in polar coordinates as

with r > 0 and , then

Ln(z) = ln(r) + i( +), where k is any integer and ln(r) is the usual natural logarithm of a real number.

A fact that the complex logarithm function is the multi-valued function explains Paradox of Bernoulli and Leibniz

The paradox of Bernoulli and Leibniz is not an “exclusive” case for the complex logarithm function. Let us look at Example 2.

Example 2

Let us consider identity

Ln(zw) = Ln(z) + Ln(w),

where z = (-+ i); w = (-1 + i).

Then

Ln(zw) = Ln[(-+i)(-1+i )] = Ln(- 4i) = Ln(4) - i

Ln(z) + Ln(w) = Ln(-+i) + Ln(-1+i ) = [Ln(2) + i] + [Ln(2) + i] =
= 2Ln(2) + i [ + ] =
= Ln(4) + i

Ln(4)
- i Ln(4) + i

An explanation of the example 2

When we deal with several properties familiar from the real logarithm we should remember that the complex logarithm is the multi-valued function.

Ln(z) = ln(|z|) + i[arg(z) +], k = 0, ±1, ±2,...

Ln(zw) = ln(4) - i + 2k1
Ln(z) + Ln(w) = ln(4) + i + 2k2
It is possible to find such k1 and k2, that
ln(4) - i + 2k1 = ln(4) + i + 2k2

For a example:

k1 = 1, k2 = 0

Ln(4) - i + 2= Ln(4) + i

Let us consider several properties of the logarithm function familiar from the real logarithm. It is necessary to remember about “” to make them always valid for the complex extension.

Real logarithm
Complex logarithm
Ln(zw) = Ln(z) + Ln(w) Ln(zw) = Ln(z) + Ln(w) + 2(k1+k2)
Ln(z/w) = Ln(z) - Ln(w) Ln(z/w) = Ln(z) - Ln(w) + 2(k1+k2)
Ln() = nLn(z) Ln() = nLn(z) + 2k
Ln() = Ln(z) Ln() = Ln(z) + 2k

k, k1, k2 are integers.

 

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
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