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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around :

Compute the elementwise values of an array using automatic threading:

Or compute the matrix LogisticSigmoid function using MatrixFunction :

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:

Plot the LogisticSigmoid [x] function:

Plot the real part of TemplateBox[{z}, LogisticSigmoid]:

Plot the imaginary part of TemplateBox[{z}, LogisticSigmoid]:

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

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]:

LogisticSigmoid is an analytic function of x:

It has no singularities or discontinuities:

LogisticSigmoid is nondecreasing:

LogisticSigmoid is injective:

LogisticSigmoid is not surjective:

LogisticSigmoid is non-negative:

LogisticSigmoid is neither convex nor concave:

TraditionalForm formatting:

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:

Compute the indefinite integral using Integrate :

Verify the anti-derivative:

Definite integral:

More integrals:

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:

LogisticSigmoid can be represented in terms of Exp :

Series representation:

LogisticSigmoid can be represented in terms of MeijerG :

LogisticSigmoid obeys the logistic differential equation :