Wacker Art Basic Functions Bild: "Kellerwald"

## Mathematical Functions - Basic Functions

### Taylor Series of the Exponential Function ### 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)

### 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)

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

### Taylor Series

Radius of convergence: |x| < 1 and x = 1. ### Definition with arctanh ### Integral of ln(x) ### Derivative of ln(x) ### Further Properties of ln(x)    ### Logarithm Functions loga(x) with Different Bases ### Logarithm loga(x) with Different Bases a

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

### Integral of ln(x) ### 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.

### Special Function ### Derivative ### Taylor Series ### 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. ### Derivative of the Logistic Function ### 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 ...

### Taylor Series ### Relations ### Inverse Hyperbolic Sine Function - Area Sinus Hyperbolicus: y = arsinh(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) ### Cube Root Function: y = x1/3

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06. März 2021 Version 2.0
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