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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.

Reciprocal complex numbers

Complex Analysis. FreeTutorial

10.1 Definition and notation of Reciprocals
      10.1.1 Modulus and argument of Reciprocals
      10.1.2 Geometric interpretation of Reciprocals

10.1 Definition and notation of Reciprocals

Any nonzero complex number z = x + yi has a unique multiplicative inverse or reciprocal 1/z such that z(1/z) = 1.
1/z = reciprocal formula. (1.16)

The reciprocal of zero is undefined. The reciprocal of the complex number z is equal to its conjugate , divided by the square of the modulus of the complex numbers z.
.      (1.17)

Example 17:
If z = 4 + 3i, then what is the reciprocal of z?
= 4 - 3i;
|z| = = = 5;
|z|2 = 25;
= (4 - 3i)/25 = 4/25 - (3/25)i.

Example 18:
If z = i, then what is the reciprocal of i?
= i;
1/i = = ((–i)/1) = i.
The reciprocal of i is its own conjugate – i.

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10.1.1 Modulus and argument of Reciprocals

We remember that the conjugate numbers have the same modulus and opposite arguments.
|z| = ||,
arg(z) = - arg().
The modulus of reciprocal of z is equal to 1/|z|. |1/z| = 1/|z|.
The argument of reciprocal is equal to argument of the conjugate number .
arg(1/z) = arg() = - arg(z).

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10.1.2 Geometric interpretation of reciprocals

The complex number z = x + yi with polar coordinates (r,) has length r > 0 and angle .
Its reciprocal 1/z with polar coordinates (1/r, - ) has length 1/r > 0 and angle - .

If r > 1, then the length of the reciprocal is 1/r < 1.
If  0 < r < 1, then 1/r < 1.

Figure 1.7 shows the reciprocal 1/z of the complex number z.

reciprocal image

Figure1.7 The reciprocal 1/z

The reciprocal 1/z of the complex number z can be visualized as its conjugate , devided by the square of the modulus of the complex numbers z.

by Tetyana Butler

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