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.
- Gudermannian can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic Examples (4)
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Exp threads elementwise over lists and matrices:
Gudermannian can be used with Interval and CenteredInterval objects:
Specific Values (3)
Find a value of for which the using Solve:
Plot the Gudermannian function:
Plot the real part of Gudermannian[x+ I y]:
Plot the imaginary part of Gudermannian[x+ I y]:
Function Properties (11)
Gudermannian is defined for all real values:
Gudermannian is defined for all complex values except branch points:
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:
Gudermannian is neither non-negative nor non-positive:
Gudermannian has no singularities or discontinuities:
Gudermannian is neither convex nor concave:
Compute the indefinite integral using Integrate:
The definite integral of Gudermannian over a period is 0:
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:
Properties & Relations (2)
Use FunctionExpand to expand Gudermannian in terms of elementary functions:
Use FullSimplify to prove identities involving the Gudermannian function:
Wolfram Research (2008), Gudermannian, Wolfram Language function, https://reference.wolfram.com/language/ref/Gudermannian.html (updated 2020).
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. Retrieved from https://reference.wolfram.com/language/ref/Gudermannian.html