gives the Gudermannian function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Gudermannian function is generically defined by .
  • Gudermannian[z] has branch cut discontinuities in the complex plane running from to for integers , where the function is continuous from the right.
  • Gudermannian can be evaluated to arbitrary numerical precision.
  • Gudermannian automatically threads over lists.


open allclose all

Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (31)

Numerical Evaluation  (5)

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:

Exp threads elementwise over lists and matrices:

Specific Values  (3)

The value at zero:

Values at infinity:

Find a value of for which the using Solve:

Substitute in the result:

Visualize the result:

Visualization  (3)

Plot the Gudermannian function:

Plot the real part of Gudermannian[x+ I y]:

Plot the imaginary part of Gudermannian[x+ I y]:

Polar plot with :

Function Properties  (5)

Gudermannian is defined for all real values:

Gudermannian is defined for all complex values except branch points:

Real range:

Gudermannian has the mirror property gd(TemplateBox[{z}, Conjugate])=TemplateBox[{{gd, (, z, )}}, Conjugate]:

Gudermannian is an odd function:

TraditionalForm formatting:

Differentiation  (3)

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the k^(th) derivative with respect to z:

Integration  (4)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

The definite integral of Gudermannian over a period is 0:

More integrals:

Series Expansions  (4)

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:

Function Representations  (4)

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:

Applications  (1)

Nonperiodic solution of a pendulum equation:

Properties & Relations  (2)

Use FunctionExpand to expand Gudermannian in terms of elementary functions:

Use FullSimplify to prove identities involving the Gudermannian function:

Introduced in 2008
Updated in 2020