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_2024_gudermannian, author="Wolfram Research", title="{Gudermannian}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Gudermannian.html}", note=[Accessed: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_gudermannian, organization={Wolfram Research}, title={Gudermannian}, year={2020}, url={https://reference.wolfram.com/language/ref/Gudermannian.html}, note=[Accessed: 10-January-2025
]}