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 the elementwise values of an array using automatic threading:

Or compute the matrix Gudermannian 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:

Find a value of for which the using Solve:

Substitute in the result:

Visualize the result:

Plot the Gudermannian function:

Plot the real part of Gudermannian[z]:

Plot the imaginary part of Gudermannian[z]:

Polar plot with :

Gudermannian is defined for all real values:

Gudermannian is defined for all complex values except branch points:

Real range:

Gudermannian has the mirror property :

Gudermannian is an odd function:

is an analytic function of for real :

It is neither analytic nor meromorphic in the complex plane:

Gudermannian is non-decreasing:

Gudermannian is injective:

Not surjective:

Gudermannian is neither non-negative nor non-positive:

Gudermannian has no singularities or discontinuities:

Gudermannian is neither convex nor concave:

TraditionalForm formatting:

The 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:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

The definite integral of Gudermannian over a period is 0:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The first-order Fourier series:

The Taylor expansion at a generic point:

Gudermannian can be applied to a power series:

Gudermannian can be represented in terms of Exp and ArcTan on the real line:

Representation as an integral on the real line:

Since Gudermannian is odd, the same result is obtained for negative :

Gudermannian can be represented in terms of Tanh and ArcTan away from the imaginary axis:

This representation is invalid on the half that is further from the origin of each branch cut strip:

Represent Gudermannian using Piecewise:

This representation is correct at all points, including branch cuts: