Gudermannian
✖
Gudermannian
Details

- 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. »
- Gudermannian can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0d4lmu5wtb48q42i-iqyvnr

Plot over a subset of the reals:

https://wolfram.com/xid/0d4lmu5wtb48q42i-hfkmiu

Plot over a subset of the complexes:

https://wolfram.com/xid/0d4lmu5wtb48q42i-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0d4lmu5wtb48q42i-d4yi7q

Scope (38)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0d4lmu5wtb48q42i-ftmrnq


https://wolfram.com/xid/0d4lmu5wtb48q42i-cbwp2i

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

https://wolfram.com/xid/0d4lmu5wtb48q42i-c09z5


https://wolfram.com/xid/0d4lmu5wtb48q42i-sf4nd

Evaluate efficiently at high precision:

https://wolfram.com/xid/0d4lmu5wtb48q42i-3j9js


https://wolfram.com/xid/0d4lmu5wtb48q42i-jdlvqt

Compute the elementwise values of an array using automatic threading:

https://wolfram.com/xid/0d4lmu5wtb48q42i-thgd2

Or compute the matrix Gudermannian function using MatrixFunction:

https://wolfram.com/xid/0d4lmu5wtb48q42i-o5jpo

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

https://wolfram.com/xid/0d4lmu5wtb48q42i-h0d6g


https://wolfram.com/xid/0d4lmu5wtb48q42i-dj6d9x

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0d4lmu5wtb48q42i-cw18bq

Specific Values (3)

https://wolfram.com/xid/0d4lmu5wtb48q42i-bmqd0y


https://wolfram.com/xid/0d4lmu5wtb48q42i-e5asej


https://wolfram.com/xid/0d4lmu5wtb48q42i-imqw0i

Find a value of for which the
using Solve:

https://wolfram.com/xid/0d4lmu5wtb48q42i-f2hrld


https://wolfram.com/xid/0d4lmu5wtb48q42i-0hv4bm


https://wolfram.com/xid/0d4lmu5wtb48q42i-ch4bjv

Visualization (3)
Plot the Gudermannian function:

https://wolfram.com/xid/0d4lmu5wtb48q42i-ecj8m7

Plot the real part of Gudermannian[z]:

https://wolfram.com/xid/0d4lmu5wtb48q42i-ouu484

Plot the imaginary part of Gudermannian[z]:

https://wolfram.com/xid/0d4lmu5wtb48q42i-clgjbv


https://wolfram.com/xid/0d4lmu5wtb48q42i-epb4bn

Function Properties (11)
Gudermannian is defined for all real values:

https://wolfram.com/xid/0d4lmu5wtb48q42i-cl7ele

Gudermannian is defined for all complex values except branch points:

https://wolfram.com/xid/0d4lmu5wtb48q42i-de3irc


https://wolfram.com/xid/0d4lmu5wtb48q42i-evf2yr


Gudermannian has the mirror property :

https://wolfram.com/xid/0d4lmu5wtb48q42i-heoddu

Gudermannian is an odd function:

https://wolfram.com/xid/0d4lmu5wtb48q42i-b9lnok

is an analytic function of
for real
:

https://wolfram.com/xid/0d4lmu5wtb48q42i-h5x4l2

It is neither analytic nor meromorphic in the complex plane:

https://wolfram.com/xid/0d4lmu5wtb48q42i-zyh0iq


https://wolfram.com/xid/0d4lmu5wtb48q42i-mn5jws

Gudermannian is non-decreasing:

https://wolfram.com/xid/0d4lmu5wtb48q42i-g6kynf

Gudermannian is injective:

https://wolfram.com/xid/0d4lmu5wtb48q42i-gi38d7


https://wolfram.com/xid/0d4lmu5wtb48q42i-je1c8


https://wolfram.com/xid/0d4lmu5wtb48q42i-hkqec4


https://wolfram.com/xid/0d4lmu5wtb48q42i-b1r9xi

Gudermannian is neither non-negative nor non-positive:

https://wolfram.com/xid/0d4lmu5wtb48q42i-84dui

Gudermannian has no singularities or discontinuities:

https://wolfram.com/xid/0d4lmu5wtb48q42i-mdtl3h


https://wolfram.com/xid/0d4lmu5wtb48q42i-f332uf

Gudermannian is neither convex nor concave:

https://wolfram.com/xid/0d4lmu5wtb48q42i-kdss3

TraditionalForm formatting:

https://wolfram.com/xid/0d4lmu5wtb48q42i-ipizt

Differentiation (3)
The first derivative with respect to z:

https://wolfram.com/xid/0d4lmu5wtb48q42i-krpoah

Higher derivatives with respect to z:

https://wolfram.com/xid/0d4lmu5wtb48q42i-z33jv

Plot the higher derivatives with respect to z:

https://wolfram.com/xid/0d4lmu5wtb48q42i-fxwmfc

Formula for the derivative with respect to z:

https://wolfram.com/xid/0d4lmu5wtb48q42i-cb5zgj

Integration (4)
Compute the indefinite integral using Integrate:

https://wolfram.com/xid/0d4lmu5wtb48q42i-bponid


https://wolfram.com/xid/0d4lmu5wtb48q42i-op9yly


https://wolfram.com/xid/0d4lmu5wtb48q42i-bfdh5d

The definite integral of Gudermannian over a period is 0:

https://wolfram.com/xid/0d4lmu5wtb48q42i-cnll80


https://wolfram.com/xid/0d4lmu5wtb48q42i-4nbst


https://wolfram.com/xid/0d4lmu5wtb48q42i-yncg8

Series Expansions (4)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0d4lmu5wtb48q42i-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0d4lmu5wtb48q42i-binhar

The first-order Fourier series:

https://wolfram.com/xid/0d4lmu5wtb48q42i-f64drv

The Taylor expansion at a generic point:

https://wolfram.com/xid/0d4lmu5wtb48q42i-jwxla7

Gudermannian can be applied to a power series:

https://wolfram.com/xid/0d4lmu5wtb48q42i-ewm016

Function Representations (4)
Gudermannian can be represented in terms of Exp and ArcTan on the real line:

https://wolfram.com/xid/0d4lmu5wtb48q42i-dc0y5j

Representation as an integral on the real line:

https://wolfram.com/xid/0d4lmu5wtb48q42i-6235u9

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

https://wolfram.com/xid/0d4lmu5wtb48q42i-pt9z62

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

https://wolfram.com/xid/0d4lmu5wtb48q42i-kl5awq


https://wolfram.com/xid/0d4lmu5wtb48q42i-li3ck7

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

https://wolfram.com/xid/0d4lmu5wtb48q42i-r0rnul

Represent Gudermannian using Piecewise:

https://wolfram.com/xid/0d4lmu5wtb48q42i-wh4lct

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

https://wolfram.com/xid/0d4lmu5wtb48q42i-rc6483

Applications (3)Sample problems that can be solved with this function
Nonperiodic solution of a pendulum equation:

https://wolfram.com/xid/0d4lmu5wtb48q42i-bluffs

https://wolfram.com/xid/0d4lmu5wtb48q42i-brkskz

Solve a differential equation with the Gudermannian function as the inhomogeneous term:

https://wolfram.com/xid/0d4lmu5wtb48q42i-cumsay

The cumulative distribution function (CDF) of the standard distribution of the hyperbolic secant is a scaled and shifted version of the Gudermannian function:

https://wolfram.com/xid/0d4lmu5wtb48q42i-bnplmf

https://wolfram.com/xid/0d4lmu5wtb48q42i-6olgr

Properties & Relations (2)Properties of the function, and connections to other functions
Use FunctionExpand to expand Gudermannian in terms of elementary functions:

https://wolfram.com/xid/0d4lmu5wtb48q42i-c2aaqc


https://wolfram.com/xid/0d4lmu5wtb48q42i-f2a1am

Use FullSimplify to prove identities involving the Gudermannian function:

https://wolfram.com/xid/0d4lmu5wtb48q42i-wmmi

Wolfram Research (2008), Gudermannian, Wolfram Language function, https://reference.wolfram.com/language/ref/Gudermannian.html (updated 2020).
Text
Wolfram Research (2008), Gudermannian, Wolfram Language function, https://reference.wolfram.com/language/ref/Gudermannian.html (updated 2020).
Wolfram Research (2008), Gudermannian, Wolfram Language function, https://reference.wolfram.com/language/ref/Gudermannian.html (updated 2020).
CMS
Wolfram Language. 2008. "Gudermannian." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/Gudermannian.html.
Wolfram Language. 2008. "Gudermannian." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/Gudermannian.html.
APA
Wolfram Language. (2008). Gudermannian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Gudermannian.html
Wolfram Language. (2008). Gudermannian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Gudermannian.html
BibTeX
@misc{reference.wolfram_2025_gudermannian, author="Wolfram Research", title="{Gudermannian}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Gudermannian.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_gudermannian, organization={Wolfram Research}, title={Gudermannian}, year={2020}, url={https://reference.wolfram.com/language/ref/Gudermannian.html}, note=[Accessed: 10-July-2025
]}