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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 Trigonometric Functions

Complex Analysis. FreeTutorial

1) Definitions
     1.1 Right triangle and unit-circle definitions
     1.2 Definitions via series
     1.3 Definitions via complex exponentials
     1.4 Definitions via differential equations

2) A difference between the real and complex trigonometric functions
     2.1 Relationship to exponential function
     2.2 The complex sine and cosine functions are not bounded

3) Identities
     3.1 Periodic identities
     3.2 Even and odd identities
     3.3 Pythagorean identity
     3.4 The sum and difference formulas
     3.5 The double-angle formulas
     3.6 More identities

4) Calculus

1) Definitions

1.1 Right triangle and unit-circle definitions

The trigonometric functions are most simply defined using the right triangle. But the right triangle definitions only define the trigonometric functions for angles between 0 and pi/2 radians.

unit circle image

The six trigonometric functions can also be defined in terms of the unit circle (the circle of radius one centered at the origin). The unit circle definition permits the definition of the trigonometric functions for all positive and negative arguments.

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1.2 Definitions via series

The complex trigonometric functions can be represented by the power series:
Sin(z) =sine as series
Cos(z) =cosine as series

Other complex trigonometric functions are:
tan(z) = tan
cot(z) = cot
sec(z) = sec
csc(z) = csc

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1.3 Definitions via complex exponentials

The complex trigonometric functions can be defined algebraically in terms of complex exponentials as:

Sin(z) = complex exponential formula sine

Cos(z) = complex exponential formula cosine

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1.4 Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation

y" = - y                              (1)

The sine function is the unique solution satisfying the initial conditions y(0) = 0 and y'(0) = 1

Let y = Sin(z)

Then y' = Cos(z); y" = - Sin(z)

Sin(z) = Sin(z)                   (1)

Initial conditions:

Sin(0) = 0, Sin'(0) = Cos(0) = 1


The cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y'(0) = 0

Let y = Cos(z)

Then y' = - Sin(z); y" = - Cos(z)

Cos(z) = Cos(z)                   (1)

Initial conditions:

Cos(0) = 1, Cos'(0) = -Sin(0) = 0


The tangent function is the unique solution of the nonlinear differential equation

y' = 1 + y2

satisfying the initial conditions y(0) = 0

Let y = tan(z)

Then y' = sec(z)2; y2 = tan(z)2

sec(z)2 = 1 + tan(z)2        (1)

It is an Pythagorean identity.

Initial conditions:

tan(0) = 0

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2) A difference between the real and complex trigonometric functions

There is a big difference between the real and complex trigonometric functions:

1) The real trigonometric functions are not related to the exponential function. But complex trigonometric functions do have Relationship to exponential function

2) The real sine and cosine functions are bounded:
|Sin(x)|
1, |Cos(x)|1

The complex sine and cosine functions are not bounded if they are defined over the set of all complex numbers.

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2.1 Relationship to exponential function

The main distinction between real and complex trigonometric functions is relationship to exponential function.

The complex trigonometric functions can be defined algebraically in terms of complex exponentials as:

Sin(z) = complex exponential formula sine

Cos(z) = complex exponential formula cosine

It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:

Euler formula
This relationship was first noted by Euler and the identity is called Euler's formula.

The relationship between the complex exponential and the trigonometric functions can be expressed as:

Sin(z) = complex sine formula= -iSinh(iz)

Cos(z) = complex cosine formula= Cosh(iz)

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2.2 The complex sine and cosine functions are not bounded

There is another distinction between real and complex trigonometric functions. In a case of complex variables

|Sin(z)|1 and |Cos(z)|1 are not true.


For example

Sin(i) = 1.17520i,
Cos(i)
= 1.54308
= -1.3811
= 2.3811

But the Pythagorean identity is true.

+= 1

+= 2.3811 - 1.3811 = 1

Example: Sin(z) = 2 Is it a paradox?

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3) Identities

3.1 Periodic identities

All complex trigonometric functions are periodic functions with the same periods as trigonometric function for real variables.
The sine, cosine, secant, and cosecant functions have period :

Sin(z +) = Sin(z)

Cos(z +) = Cos(z)

sec(z +) = sec(z)

csc(z +) = csc(z)

The tangent and cotangent functions have period :

tan(z +) = tan(z)

cot(z +) = cot(z)

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3.2 Even and odd functions

Cos(z) is an even function, Sin(z) is an odd function as trigonometric functions for real variables.

Sin(-z) = - Sin(z)

Cos(-z) = Cos(z)

sec(-z) = sec(z)

csc(-z) = - csc(z)

tan(-z) = - tan(z)

cot(-z) = - cot(z)

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3.3 Pythagorean identities

These identities are based on the Pythagorean theorem.

+= 1

The second equation is obtained from the first by dividing both sides by .

tan(z)2 + 1 = sec(z)2

The third equation is obtained from the first by dividing both sides by .

1 + cot(z)2 = csc(z)2

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3.4 The sum and difference formulas

Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves.

Sin(z1+z2) = Sin(z1)Cos(z2) + Cos(z1)Sin(z2)

Sin(z1-z2) = Sin(z1)Cos(z2) - Cos(z1)Sin(z2)

Cos(z1+z2) = Cos(z1)Cos(z2) - Sin(z1)Sin(z2)

Cos(z1-z2) = Cos(z1)Cos(z2) + Sin(z1)Sin(z2)

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulas.

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3.5 The double-angle formulas

Sin(2z) = 2Sin(z)Cos(z)

Cos(2z) = -

3.6 More identities

Sin(z +) = - Sin(z)

Cos(z +) = - Cos(z)

Cos(z) = Sin(z + )

Sin( + z) = Sin( - z)

The complex modulus satisfies modulus identity:

|Sin(x + iy)| = |Sin(x) + Sin(iy)|

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4) Calculus

If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.

derivativeSin(z) = Cos(z)

derivativeCos(z) = - Sin(z)

The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation.

derivativetan(z) = sec(z)2

derivativecot(z) = - csc(z)2

derivativesec(z) = sec(z)tan(z)

derivativecsc(z) = - csc(z)cot(z)

 

by Tetyana Butler

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