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:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix InverseGudermannian function using MatrixFunction:

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

Or compute average-case statistical intervals using Around:

The value at zero:

Values at infinity:

Exact evaluation:

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

Plot the InverseGudermannian function:

Plot the real part of InverseGudermannian:

Plot the imaginary part of InverseGudermannian:

Polar plot with :

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:

TraditionalForm formatting:

The first derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x:

Compute the indefinite integral using Integrate:

The definite integral:

The definite integral of InverseGudermannian over a period is 0:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

FourierSeries:

InverseGudermannian can be applied to a power series: