# InverseGudermannian

gives the inverse Gudermannian function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The inverse Gudermannian function is defined by .
• has branch cut discontinuities in the complex plane running from to for integers .
• InverseGudermannian can be evaluated to arbitrary numerical precision.
• InverseGudermannian 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:

Asymptotic expansion at a singular point:

## Scope(30)

### Numerical Evaluation(5)

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:

### 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: