Wacker Art Basic Functions

Wacker Art

Basic Functions

Bild: "Kellerwald"

Mathematical Functions - Basic Functions

Exponential Function: y = e^x

Taylor Series of the Exponential Function

Exponential Function Taylor Series

Even Part of the Exponential Function - The Hyperbolic Cosine

Cosinus Hyperbolicus Function

A function is called even if f(x) = f(-x)

Odd Part of the Exponential Function - The Hyperbolic Sine

Hyperbolic Sine Function

A function is called odd if f(x) = -f(-x)

Taylor Series of the Hyperbolic Cosine Function

Hyperbolic Cosine Function Taylor Series

Taylor Series of the Hyperbolic Sine Function

Hyperbolic Sine Function Taylor Series

Relation between Hyperbolic Cosine Function and Cosine Function

Cosine Function

Relation between Hyperbolic Sine Function and Sine Function

Sine Function

Taylor Series of the Cosine Function

Cosine Function Taylor Series

Taylor Series of the Sine Function

Sine Function Taylor Series

Basic Relations

Exponential Function

Hyperbolic Functions Square Basic Relations

Trigonometric Functions Square Basic Relations

Definition of Sine and Cosine Functions with the Complex Exponential Function

Cosine Function

Sine Function

Definition of Tangent and Cotangent Functions with Sine and Cosine Functions

Tanget Function

Cotangent Function

Hyperbolic Tangent and Hyperbolic Cotangent Functions

Hyperbolic Tangent Function

Hyperbolic Cotangent Function

Logarithm Function: y = ln(x)

The logarithm function is defined for x values in the range 0 < x < ∞.

Taylor Series

Radius of convergence: |x| < 1 and x = 1.

Logarithem Function

Definition with arctanh

Logarithem Function and arctanh

Integral of ln(x)

Intergral of ln(x)

Derivative of ln(x)

Further Properties of ln(x)

Logarithem Sum

Logarithem Difference

Logarithem of a power b

Logarithem of a root n

Logarithm Functions log_a(x) with Different Bases

Logarithem of Base a

Logarithm Base 10 Function : y = log₁₀(x)

Logarithm Base 1/2 Function : y = log_1/2(x)

Logarithm Base 2 Function : y = log₂(x)

Logarithm log_a(x) with Different Bases a

a = 1/2
a = 2
a = e
a = 10

Integral of the Logarithm Function: y = ln(x)x -x

Integral of ln(x)

Intergral of ln(x)

Function: y = x/ln(x)

Function

xln(x)

This function can be used for the approximation of the prime counting function π(x).

xln(x)

This relation is also known as prime number theorem.

Prime Number Theorem

Special Function

y = x^x = e^ln(x)x

Derivative

Special Function Derivative

y = e^ln(x)x(ln(x)+1)

Gauß Function: y = e^-x²

Taylor Series

Gauss Function Taylor Series

Function: y = e^-1/x²

Logistic Function: y = 1/(1 + e^-x)

Formulars of the logistic function:

Logistic Function

The logistic function is the solution of the logistic differential equation.

Logistic Differential Equation

Integration of the Logistic Function

Using the substitution u = 1 + e^x and u' = e^x gives du = e^xdx.

Logistic Function

Integral of the Logistic Function: y = ln(1+e^x)

Derivative of the Logistic Function

Logistic Function

Derivative of the Logistic Function: y = e^x/(1+e^x)²

Bernoulli Function: x/(1 - e^-x)

Bernoulli Function: x/(e^-x - 1)

Bernoulli Function

Hyperbolic Sine Function Taylor Series

The B_i are the Bernoulli Numbers.

Bernoulli Numbers

The Bernoulli numbers B_i and Bernoulli numbers B_i^* differ only in the value of i= 1. B₁=-1/2 and B₁^* = 1/2, where they only have different sign. All the other values are the same.

i	0	1	2	3	4	5	6	7	8	9	10	11	12	...
B_i	1	±1/2	1/6	0	-1/30	0	1/42	0	-1/30	0	5/66	0	-291/2730	...

Hyperbolic Sine Function - Sinus Hyperbolicus: y = sinh(x)

Taylor Series

Hyperbolic Sine Function Taylor Series

Relations

Hyperbolic Sine Function

Inverse Hyperbolic Sine Function - Area Sinus Hyperbolicus: y = arsinh(x)

Hyperbolic arsine Function

Hyperbolic Cosine Function - Cosinus Hyperbolicus: y = cosh(x)

Taylor Series

Hyperbolic Cosine Function Taylor Series

Relations

Cosinus Hyperbolicus Function

Invers Hyperbolic Cosine Function - Area Cosinus Hyperbolicus: y = arcosh(x)

Inverse hyperbolic cosine Function

Hyperbolic Tangent Function - Tangens Hyperbolicus: y = tanh(x)

Hyperbolic Tangent Function

Inverse Hyperbolic Tangent Function - Area Tangens Hyperbolicus: y = artanh(x)

Invers Hyperbolic Tangent Function

Hyperbolic Cotangent Function - Cotangens Hyperbolicus: y = coth(x)

Hyperbolic Cotangent Function

Inverse Hyperbolic Cotangent Function - Area Cotangens Hyperbolicus: y = arcoth(x)

Invers Hyperbolic Cotangent Function

Sine Function: y = sin(x)

Sine Function Taylor Series

Inverse Sine Function - Arcus Sine: y = arcsin(x)

arcsine Function Taylor Series

Cosine Function: y = cos(x)

Cosine Function Taylor Series

Inverse Cosine Function - Arcus Cosine: y = arccos(x)

Arc Cosine Function Taylor Series

Tangent Function: y = tan(x)

Tangent Function

Inverse Tangent Function: y = arctan(x)

Arcus Tangent Function

Cotangent Function: y = cot(x)

Cotangent Function

Invers Cotangent Function: y = arccot(x)

Arcus Cotangent Function

Power Functions: y = xⁿ

y = x^-4	y = x^-3
y = x^-2	y = x^-1
y = x⁰ = 1	y = x¹ = x
y = x²	y = x³
y = x⁴	y = x⁵
y = x²⁴	y = x²⁵

Square Root Function: y = x^1/2

Cube Root Function: y = x^1/3

Bild: "Edersee"

Further Functions

06. März 2021 Version 2.0

Mathematical Functions - Basic Functions

Exponential Function: y = ex

Taylor Series of the Exponential Function

Even Part of the Exponential Function - The Hyperbolic Cosine

Odd Part of the Exponential Function - The Hyperbolic Sine

Taylor Series of the Hyperbolic Cosine Function

Taylor Series of the Hyperbolic Sine Function

Relation between Hyperbolic Cosine Function and Cosine Function

Relation between Hyperbolic Sine Function and Sine Function

Taylor Series of the Cosine Function

Taylor Series of the Sine Function

Basic Relations

Definition of Sine and Cosine Functions with the Complex Exponential Function

Definition of Tangent and Cotangent Functions with Sine and Cosine Functions

Hyperbolic Tangent and Hyperbolic Cotangent Functions

Logarithm Function: y = ln(x)

Taylor Series

Definition with arctanh

Integral of ln(x)

Derivative of ln(x)

Further Properties of ln(x)

Logarithm Functions loga(x) with Different Bases

Logarithm Base 10 Function : y = log10(x)

Logarithm Base 1/2 Function : y = log1/2(x)

Logarithm Base 2 Function : y = log2(x)

Logarithm loga(x) with Different Bases a

Integral of the Logarithm Function: y = ln(x)x -x

Integral of ln(x)

Function: y = x/ln(x)

Function

Special Function

y = xx = eln(x)x

Derivative

y = eln(x)x(ln(x)+1)

Gauß Function: y = e-x2

Taylor Series

Function: y = e-1/x2

Logistic Function: y = 1/(1 + e-x)

Integration of the Logistic Function

Integral of the Logistic Function: y = ln(1+ex)

Derivative of the Logistic Function

Derivative of the Logistic Function: y = ex/(1+ex)2

Bernoulli Function: x/(1 - e-x)

Bernoulli Function: x/(e-x - 1)

Bernoulli Function

Bernoulli Numbers

Hyperbolic Sine Function - Sinus Hyperbolicus: y = sinh(x)

Taylor Series

Relations

Inverse Hyperbolic Sine Function - Area Sinus Hyperbolicus: y = arsinh(x)

Hyperbolic Cosine Function - Cosinus Hyperbolicus: y = cosh(x)

Taylor Series

Relations

Invers Hyperbolic Cosine Function - Area Cosinus Hyperbolicus: y = arcosh(x)

Hyperbolic Tangent Function - Tangens Hyperbolicus: y = tanh(x)

Inverse Hyperbolic Tangent Function - Area Tangens Hyperbolicus: y = artanh(x)

Hyperbolic Cotangent Function - Cotangens Hyperbolicus: y = coth(x)

Inverse Hyperbolic Cotangent Function - Area Cotangens Hyperbolicus: y = arcoth(x)

Sine Function: y = sin(x)

Inverse Sine Function - Arcus Sine: y = arcsin(x)

Cosine Function: y = cos(x)

Inverse Cosine Function - Arcus Cosine: y = arccos(x)

Tangent Function: y = tan(x)

Inverse Tangent Function: y = arctan(x)

Cotangent Function: y = cot(x)

Invers Cotangent Function: y = arccot(x)

Power Functions: y = xn

y = x-4

y = x-3

y = x-2

y = x-1

y = x0 = 1

y = x1 = x

y = x2

y = x3

y = x4

y = x5

y = x24

y = x25

Square Root Function: y = x1/2

Exponential Function: y = e^x

Logarithm Functions log_a(x) with Different Bases

Logarithm Base 10 Function : y = log₁₀(x)

Logarithm Base 1/2 Function : y = log_1/2(x)

Logarithm Base 2 Function : y = log₂(x)

Logarithm log_a(x) with Different Bases a

y = x^x = e^ln(x)x

y = e^ln(x)x(ln(x)+1)

Gauß Function: y = e^-x²

Function: y = e^-1/x²

Logistic Function: y = 1/(1 + e^-x)

Integral of the Logistic Function: y = ln(1+e^x)

Derivative of the Logistic Function: y = e^x/(1+e^x)²

Bernoulli Function: x/(1 - e^-x)

Bernoulli Function: x/(e^-x - 1)

Power Functions: y = xⁿ

y = x^-4

y = x^-3

y = x^-2

y = x^-1

y = x⁰ = 1

y = x¹ = x

y = x²

y = x³

y = x⁴

y = x⁵

y = x²⁴

y = x²⁵

Square Root Function: y = x^1/2

Cube Root Function: y = x^1/3