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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 Trigonometric Function

Complex Analysis. FreeTutorial

1) The notations

2) Range of usual principal value

3) Definitions as infinite series

4) Logarithmic forms
     4.1 Natural logarithm's expressions
     4.2 Logarithmic formulas and connections

     4.3 Logarithmic formulas. Proofs

5) Derivatives of inverse trigonometric functions

6) Indefinite integrals of inverse trigonometric functions

1) The notations

For inverse trigonometric functions, the notations sin-1 and cos-1 are often used for arcsin and arccos, 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 trigonometric and hyperbolic functions are the multivalued function that are the inverse functions of the trigonometric and hyperbolic functions.

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2) Range of usual principal value

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse.

y = f(z) Alternate notations Range of usual principal value
y=sin-1(z) y=arcsin(z)
y=cos-1(z) y=arccos(z)
y=tan-1(z) y=arctan(z)
y=cot-1(z) y=arccot(z)
y
0
y=sec-1(z) y=arcsec(z)
y
y=csc-1(z) y=arccsc(z)
y
0

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3) Definitions as infinite series

The inverse trigonometric functions can be defined in terms of infinite series.

arcsin(z)=arcsin series

arccos(z)= - arcsin(z) =

= - arccos series

arctan(z)=arctan series

arccot(z)= - arctan(z) =

= - arccot series

arccsc(z) = arcsin(z -1) =

= arccsc series

arcsec(z) = arccos(z -1) =

= - arcsec series

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4) Logarithmic forms

4.1 Natural logarithm's expressions

The inverse trigonometric functions may be expressed using natural logarithms.

arcsin(z) = -i ln(iz +)

arccos(z) = -i ln(z +)

arctan(z) = (ln(1 - iz) - (ln(1 + iz))

arccot(z) = (ln(1 - ) - (ln(1 + ))

arccsc(z) = -i ln(+ )

arcsec(z) = -i ln(+ )

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4.2 Logarithmic formulas and connections

arcsin(z) = arccsc( 4.3.1 Proof
arccos(z) = arcsec() 4.3.2 Proof
arctan(z) = arccot() 4.3.3 Proof
arccot(z) = arctan() 4.3.4 Proof
arccsc(z) =  arcsin( 4.3.5 Proof
arcsec(z) = arccos() 4.3.6 Proof

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4.3 Logarithmic formulas and connections. Proofs

4.3.1 arcsin(z) = -i ln(iz +) =
= arccsc()
4.3.2

arccos(z) = -i ln(z +) =

= + i ln(iz +) =

= - arcsin(z) = arcsec()

4.3.3

arctan(z) = (ln(1 - iz) - (ln(1 + iz)) = arccot()

4.3.4

arccot(z) = (ln(1 - ) - (ln(1 + )) = arctan()

4.3.5

arccsc(z) = -i ln(+ ) =

= arcsin()

4.3.6

arcsec(z) = -i ln(+ ) =

= i ln(+ ) + =

= - arccsc(z) = arccos()

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5) Derivatives of inverse trigonometric functions

derivativearcsin(z) =

derivativearccos(z) = -

derivativearctan(z) =

derivativearccot(z) = -

arccsc(z) = -

derivativearcsec(z) =

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6) Indefinite integrals of inverse trigonometric functions

integralarcsin(z)dz = zarcsin(z) + + C

integralarccos(z)dz = zarccos(z) - + C

integralarctan(z)dz = zarctan(z) - ln(1+ z2) + C

integralarccot(z)dz = zarccot(z) + ln(1+ z2) + C

integralarccsc(z)dz = zarccsc(z) + ln(z + ) + C

integralarcsec(z)dz = zarcsec(z) - ln(z + ) + C

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
Set theory paradoxes
We will add more
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