gives the inverse Gudermannian function .


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


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at a singular point:

Scope  (22)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

InverseGudermannian threads elementwise over lists and matrices:

Specific Values  (4)

The value at zero:

Values at infinity:

Exact evaluation:

Find a value of x for which the InverseGudermannian[x]=0.8 using Solve:

Visualization  (3)

Plot the InverseGudermannian function:

Plot the real part of InverseGudermannian:

Plot the imaginary part of InverseGudermannian:

Polar plot with :

Function Properties  (2)

InverseGudermannian is defined on disjoint intervals of real axis:

InverseGudermannian is defined for all integer complex values:

TraditionalForm formatting:

Differentiation  (2)

The first derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x:

Integration  (3)

Compute the indefinite integral using Integrate:

The definite integral:

The definite integral of InverseGudermannian over a period is 0:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :


InverseGudermannian can be applied to a power series:

Applications  (1)

Mercator projection map of the world:

Properties & Relations  (2)

Derivative of the inverse Gudermannian function:

Use FunctionExpand to expand InverseGudermannian in terms of elementary functions:

Wolfram Research (2008), InverseGudermannian, Wolfram Language function,


Wolfram Research (2008), InverseGudermannian, Wolfram Language function,


@misc{reference.wolfram_2020_inversegudermannian, author="Wolfram Research", title="{InverseGudermannian}", year="2008", howpublished="\url{}", note=[Accessed: 19-April-2021 ]}


@online{reference.wolfram_2020_inversegudermannian, organization={Wolfram Research}, title={InverseGudermannian}, year={2008}, url={}, note=[Accessed: 19-April-2021 ]}


Wolfram Language. 2008. "InverseGudermannian." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2008). InverseGudermannian. Wolfram Language & System Documentation Center. Retrieved from