LogisticSigmoid

LogisticSigmoid[z]

gives the logistic sigmoid function.

Details

  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • In TraditionalForm, the logistic sigmoid function is sometimes denoted as .
  • The logistic function is a solution to the differential equation .
  • LogisticSigmoid[z] has no branch cut discontinuities.
  • LogisticSigmoid can be evaluated to arbitrary numerical precision.
  • LogisticSigmoid automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

The expansion of the function:

Scope  (29)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

LogisticSigmoid threads elementwise over lists and matrices:

Specific Values  (4)

The value of LogisticSigmoid at 2 πI n for integer n is 1/2:

Values at infinity:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Find a value of for which the TemplateBox[{x}, LogisticSigmoid]=0.8` using Solve:

Substitute in the result:

Visualize the result:

Visualization  (3)

Plot the LogisticSigmoid[x] function:

Plot the real part of TemplateBox[{{x, +, {i,  , y}}}, LogisticSigmoid]:

Plot the imaginary part of TemplateBox[{{x, +, {i,  , y}}}, LogisticSigmoid]:

Polar plot with TemplateBox[{phi}, LogisticSigmoid]:

Function Properties  (4)

LogisticSigmoid is defined for all real and complex values:

LogisticSigmoid achieves all values between 0 and 1 on the reals:

The range for complex values:

LogisticSigmoid has the mirror property TemplateBox[{{z, }}, LogisticSigmoid]=TemplateBox[{z}, LogisticSigmoid]:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the ^(th) derivative with respect to z:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Taylor expansion at a generic point:

Function Representations  (4)

LogisticSigmoid can be represented in terms of Exp:

Series representation:

LogisticSigmoid can be represented in terms of MeijerG:

LogisticSigmoid obeys the logistic differential equation :

Introduced in 2014
 (10.0)