# LogisticSigmoid

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 .
• has no branch cut discontinuities.
• LogisticSigmoid can be evaluated to arbitrary numerical precision.
• LogisticSigmoid automatically threads over lists.
• LogisticSigmoid can be used with Interval and CenteredInterval objects. »

# 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(36)

### Numerical Evaluation(6)

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:

LogisticSigmoid can be used with Interval and CenteredInterval objects:

### 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 using Solve:

Substitute in the result:

Visualize the result:

### Visualization(3)

Plot the function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(10)

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 :

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:

### 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 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 :

## Applications(1)

Write a specific solution to the dimensionless logistic equation using LogisticSigmoid:

Wolfram Research (2014), LogisticSigmoid, Wolfram Language function, https://reference.wolfram.com/language/ref/LogisticSigmoid.html.

#### Text

Wolfram Research (2014), LogisticSigmoid, Wolfram Language function, https://reference.wolfram.com/language/ref/LogisticSigmoid.html.

#### CMS

Wolfram Language. 2014. "LogisticSigmoid." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LogisticSigmoid.html.

#### APA

Wolfram Language. (2014). LogisticSigmoid. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LogisticSigmoid.html

#### BibTeX

@misc{reference.wolfram_2024_logisticsigmoid, author="Wolfram Research", title="{LogisticSigmoid}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/LogisticSigmoid.html}", note=[Accessed: 24-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_logisticsigmoid, organization={Wolfram Research}, title={LogisticSigmoid}, year={2014}, url={https://reference.wolfram.com/language/ref/LogisticSigmoid.html}, note=[Accessed: 24-July-2024 ]}