Wacker Art Basic Functions
Bild: "Kellerwald"

## Mathematical Functions - Basic Functions

### Even Part of the Exponential Function - The Hyperbolic Cosine

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

### Odd Part of the Exponential Function - The Hyperbolic Sine

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

### 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.

### Logarithm loga(x) with Different Bases a

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

### Function

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

This relation is also known as prime number theorem.

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

Formulars of the logistic function:

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

### Integration of the Logistic Function

Using the substitution u = 1 + ex and u' = ex gives du = exdx.

### Bernoulli Function

The Bi are the Bernoulli Numbers.

### Bernoulli Numbers

The Bernoulli numbers Bi and Bernoulli numbers Bi* 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 ... Bi 1 ±1/2 1/6 0 -1/30 0 1/42 0 -1/30 0 5/66 0 -291/2730 ...

### Cube Root Function: y = x1/3

Bild: "Edersee"

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06. März 2021 Version 2.0
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