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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.

A geometric interpretation of division

Complex Analysis. FreeTutorial

11.1 Division as multiplication and reciprocation

11.2 The modulus and argument of the quotient
      11.2.1 Example of division


11.1 Division as multiplication and reciprocation

Let us replace division z1/z2 with multiplication z1(1/z2), where 1/z2 is the reciprocal of z2.

Step1: Construct conjugate 2 = (x2, -y2).
Step2: Construct reciprocal .
Step3: Perform multiplication z1(1/z2) = z1/z2.

Such way the division can be compounded from multiplication and reciprocation. Figure 1.18 shows all steps.

division of the complex  numbers image
Figure 1.18 Division of the complex numbers z1/z2


11.2 The modulus and argument of the quotient

Let us consider two complex numbers z1 and z2 in a polar form.
z1 = r1(cos+i sin),
z2 = r2(cos+i sin).
It was shown (1.18) that their product z1z2 has the modulus r1r2 and the argument +.
We have replaced division z1/z2 with multiplication z1(1/z2).
Let 1/z2 = z3 .
z3 = r3(cos+i sin).
z1/z2 = z1z3 = r1r3[(cos(+) + i(sin(+)].
The modulus r3 is the modulus of reciprocal 1/z2.
r3 = |1/z2|.
The argument is the argument of reciprocal 1/z2.
= arg(1/z2) = - arg(z2).

The quotient z1/z2 has the modulus |z1|/|z2| and the argument {Arg(z1) - Arg(z2)}.
|z1/z2| = |z1|/|z2|,        (1.21)
Arg(z1/z2) = Arg(z1) - Arg(z2).      (1.22)


11.2.1 Example of division

z1 = 2 + i,
2= 2 + i.
Find modulus |z1/z2| and arg(z1/z2).
Construct z1/z2 using multiplication and reciprocation.

|z1| = 4,
|z2| = ,
|z1/z2| = |z1|/|z2| = .
arg(z1) = 60º,
arg(z2) = 30º.
arg(z1/z2) = arg(z1) - arg(z2) = 60º - 30º= 30º
z1/z2 = (cos(30º)+ i sin(30º)).

Let us construct z1/z2 using multiplication and reciprocation.
Step1: Construct conjugate 2.
|2| = , arg(2) = -30º.
Step2: Calculate |z2|2 and construct reciprocal of z2.
|z2|2 = ()2 = 16/3,
Reciprocal has polar coordinates (, -30º).
Step3: Perform multiplication z1(1/z2) = z1/z2.
z1(1/z2) = (4, 60º)(, -30º) = (, 30º).
z1/z2 = (cos(30º)+ i sin(30º)).
See Figure1.19.

example of division image
Figure 1.19 Example z1/z2

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
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We will add more
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