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

Next Page:

06. März 2021 Version 2.0
Copyright: Hermann Wacker Uhlandstraße 10 D-85386 Eching bei Freising Germany Haftungsausschluß