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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 addition and subtraction of complex numbers

Complex Analysis. FreeTutorial

6.1 A geometric interpretation of addition

6.2 A geometric interpretation of subtraction

6.1 A geometric interpretation of addition

Geometrically, addition of two complex numbers z1 and z2 can be visualized as addition of the vectors by using the "parallelogram law". The vector sum z1 + z2 is represented by the diagonal of the parallelogram formed by the two original vectors.

Addidion image
Figure 1.8 Addition of two complex numbers

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6.2 A geometric interpretation of subtraction

Figure 1.9 shows that the difference z1 - z2 is represented by the vector joining the point z2 to the point z1.

subractiom image 1

Figure 1.9 The difference z1 - z2

The easiest way to represent the difference z1 - z2 is to think in terms of adding a negative vector z1 + (- z2). The negative vector is the same vector as its positive counterpart, only pointing in the opposite direction. See Figure 1.10

nagative vector image

Figure 1.10 The negative vector

Now we have to perform vector addition z1 + (- z2). See Figure1.11 and Figure1.12.


negative vector image 2

Figure1.11 Negative vector (- z2)

subractiom image 2

Figure1.12 z1 - z2 = z1 + (- z2)

Note that the difference vector z1 - z2 may be drawn from the tip of z2 to the tip of z1 rather than from the origin. This is a common practice which emphasizes relationships among vectors, but the translation in the plot has no effect on the mathematical definition or properties of the vector. See Figure1.13

subractiom image 3

Figure1.13 z1 - z2

Subtraction is not commutative. z1 - z2 z2z1. Compare Figure 1.13 and 1.14.

subractiom image 4
Figure 1.14 z2z1

The difference z1 - z2 is represented by the vector joining the point z2 to the point z1.
The difference z2z1 is represented by the vector joining the point z1 to the point z2.

by Tetyana Butler

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