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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 Inverse Hyperbolic Functions

Complex Analysis. FreeTutorial

1) The notations

2) Definitions    

3) Derivatives of inverse hyperbolic functions

4) Indefinite integrals

1) The notations

For inverse hyperbolic functions, the notations sinh-1 and cosh-1 are often used for arcsinh and arccosh, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion.

The inverse hyperbolic functions are the multivalued function that are the inverse functions of the hyperbolic functions.

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2) Definitions

The inverse hyperbolic functions may be expressed using natural logarithms.

arcsinh(z) = ln(z + )

arccosh(z) = ln()
For real x > 1, this simplifies to
arccosh(x) = ln(x + )

arctanh(z) = [ln(1 + z) - ln(1 - z)]
For real x < 1, this simplifies to
arctanh(x) = ln()

arccoth(z) = [ln(1 + ) - ln(1 - )]
For real x < 0 or x >1, this simplifies to
arccoth(x) = ln()

arccsch(z) = ln()
For real x, it satisfies
arccsch(x)
=arccsch r

arcsech(z) = ln()
For real x, it satisfies
arcsech(x)
=arcsech


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3) Derivatives of inverse hyperbolic functions

derivativearcsinh(z) = derivative arcsinh

derivativearccosh(z) = derivative arccosh

derivativearctanh(z) = derivative arctanh

derivativearccoth(z) = derivative arccoth

derivativearccsch(z) = - derivative arccsch

derivativearcsech(z) = - derivative arcsech

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4) Indefinite integrals

integralsinh-1zdz = zsinh-1z - + C

integralcosh-1zdz = zcosh-1z - (z+1)+ C

integraltanh-1zdz = ztanh-1z + ln(z2 - 1) + C
integralcoth-1zdz = zcoth-1z + ln(z2 - 1) + C
integralcsch-1zdz = zcsch-1z + ln(z (1 + )) + C
integralsech-1zdz = zsech-1z - tan-1() + C

 

by Tetyana Butler

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