# InverseGudermannian

gives the inverse Gudermannian function .

# 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:

Asymptotic expansion at a singular point:

## Scope(31)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

InverseGudermannian threads elementwise over lists and matrices:

InverseGudermannian can be used with Interval and CenteredInterval objects:

### Specific Values(4)

The value at zero:

Values at infinity:

Exact evaluation:

Find a value of x for which the InverseGudermannian[x]=0.8 using Solve:

### Visualization(3)

Plot the InverseGudermannian function:

Plot the real part of InverseGudermannian:

Plot the imaginary part of InverseGudermannian:

Polar plot with :

### Function Properties(10)

InverseGudermannian is defined on disjoint intervals of real axis:

InverseGudermannian is defined for all integer complex values:

InverseGudermannian achieves all real values:

InverseGudermannian is not an analytic function:

Nor is it meromorphic:

InverseGudermannian is neither non-decreasing nor non-increasing:

InverseGudermannian is not injective:

InverseGudermannian is surjective:

InverseGudermannian is neither non-negative nor non-positive:

InverseGudermannian has both singularity and discontinuity in [π/2, 3π/2]:

InverseGudermannian is neither convex nor concave:

### Differentiation(2)

The first derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x:

### Integration(3)

Compute the indefinite integral using Integrate:

The definite integral:

The definite integral of InverseGudermannian over a period is 0:

More integrals:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

InverseGudermannian can be applied to a power series:

## Applications(2)

Mercator projection map of the world:

Solve a differential equation with the inverse Gudermannian function as the inhomogeneous part:

## Properties & Relations(2)

Derivative of the inverse Gudermannian function:

Use FunctionExpand to expand InverseGudermannian in terms of elementary functions:

Wolfram Research (2008), InverseGudermannian, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseGudermannian.html.

#### Text

Wolfram Research (2008), InverseGudermannian, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseGudermannian.html.

#### CMS

Wolfram Language. 2008. "InverseGudermannian." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseGudermannian.html.

#### APA

Wolfram Language. (2008). InverseGudermannian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseGudermannian.html

#### BibTeX

@misc{reference.wolfram_2024_inversegudermannian, author="Wolfram Research", title="{InverseGudermannian}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/InverseGudermannian.html}", note=[Accessed: 12-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_inversegudermannian, organization={Wolfram Research}, title={InverseGudermannian}, year={2008}, url={https://reference.wolfram.com/language/ref/InverseGudermannian.html}, note=[Accessed: 12-July-2024 ]}