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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.

Trigonometric and algebraic form conversion

Complex Analysis. FreeTutorial

4.1 Conversion from trigonometric to algebraic form
4.2 Conversion from algebraic to trigonometric form
4.3 Examples of the conversion from algebraic to
trigonometric form

4.1 The conversion from trigonometric to algebraic form of the complex number

The conversion from trigonometric to algebraic form of the complex number is given by
x
= |z|cos arg(z),
y
= |z|sin arg(z).                         (1.6)

Example 3:
Trigonometric form:
z = 4(cos+ i sin).
Algebraic form:
x = 4cos = 2,
y = 4sin = 2,
z = 2 + 2i.

Top

4.2 The conversion from algebraic to trigonometric form of the complex number

The conversion from algebraic to trigonometric form of the complex number is given by
|z| = square (x**2 + y**2),

arg (z) image (1.7)

It is necessary to be careful in specifying the choices of so that the point z lies in the appropriate quadrant.

Example 4:
Let us consider 4 cases, when the point z lies in 4 different quadrants. Let Re(z) and Im(z) = ±1.

z

arg(z)

Quadrant 1

quadrant 1 image

Formula for arg(z)
arg(z) =arctan image; x>0, y0
Result for arg(z)
arg(z) = arctan(1) =

Quadrant 2

quadrant 2 image

Formula for arg(z)
arg(z) =arctan image +pi; x<0, y>0
Result for arg(z)
arg(z) =arctan(-1)+pi= -+ =

Quadrant 3

quadrant 3 image

Formula for arg(z)
arg(z) =arctan image - pi; x<0, y<0
Result for arg(z)
arg(z) =arctan(1)+pi==

Quadrant 4

quadrant 4 image

Formula for arg(z)
arg(z) =arctan image; x>0, y0
Result for arg(z)
arg(z) = arctan(-1) = -

The trigonometric form of z:
z = |z|(cos Arg(z) +i sin Arg(z)).
Re(z) and Im(z) = ±1. |z| = square (x**2 + y**2)= 2 for all cases.

z = 1 +  i z = (cos + i sin)
z = -1 +  i z = (cos+ i sin)
z = -1 -  i z = (cos() + i sin())
z = 1 -  i z = (cos(-) + i sin(-))

See Figure 1.5 for all 4 cases of this example.

arg (z) examples image

Figure 1.5 Example of calculating arg(z); the point z lies in different quadrants.

You can see more examples of the conversion from algebraic to trigonometric form. Examples conversion

If z is purely imaginary number, then x = 0, and becomes undefined. We emphasize these special cases:

arg (z) image (1.8)

Example 5:
z = i
What is the trigonometric form of z?
z = |z|(cos arg(z) +i sin arg(z)).
|z| = square (x**2 + y**2)= 1
arg(z) = pi/2 gif because x = 0, y >0.

z = cospi/2 gif + i sinpi/2 gif

If y = 0, then z becomes a real number.

arg (z) images (1.9)

Example 6:
z = 1
What is the trigonometric form of z?
z = |z|(cos arg(z) +i sin arg(z)).
|z| = 1
arg(z) = 0 because x >0, y =0.
z
= cos(0) + i sin(0) = 1.

The algebraic form coincides with trigonometric one for real numbers.

arg(z) is indeterminate if x = 0, y = 0.

In selecting the proper values for arg(z), we must be careful in specifying the choices of so that the point z lies in the appropriate quadrant.

Examples of the conversion from algebraic to trigonometric form

by Tetyana Butler

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