gives the hyperbolic cosine integral TemplateBox[{z}, CoshIntegral].


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{z}, CoshIntegral]=gamma+log(z)+int_0^z(cosh(t)-1)/tdt, where is Eulers constant.
  • CoshIntegral[z] has a branch cut discontinuity in the complex z plane running from - to 0.
  • For certain special arguments, CoshIntegral automatically evaluates to exact values.
  • CoshIntegral can be evaluated to arbitrary numerical precision.
  • CoshIntegral automatically threads over lists.
  • CoshIntegral can be used with Interval and CenteredInterval objects. »


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Basic Examples  (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (37)

Numerical Evaluation  (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate CoshIntegral efficiently at high precision:

CoshIntegral threads elementwise over lists:

CoshIntegral can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Value at the origin:

Values at infinity:

Find the zero of CoshIntegral:

Visualization  (2)

Plot the CoshIntegral function:

Plot the real part of TemplateBox[{z}, CoshIntegral]:

Plot the imaginary part of TemplateBox[{z}, CoshIntegral]:

Function Properties  (9)

CoshIntegral is defined for all real positive values:

Complex domain:

CoshIntegral takes all the real values:

CoshIntegral is not an analytic function:

Nor is it meromorphic:

CoshIntegral is increasing on its real domain:

CoshIntegral is injective:

CoshIntegral is surjective:

CoshIntegral is neither non-negative nor non-positive:

It has both singularity and discontinuity in (-,0]:

CoshIntegral is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of CoshIntegral:

Definite integral:

More integrals:

Series Expansions  (3)

Series expansion for CoshIntegral:

Plot the first three approximations for CoshIntegral around :

Find asymptotic series expansion at infinity:

CoshIntegral can be applied to power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:


Function Identities and Simplifications  (3)

Primary definition of CoshIntegral:

Argument simplifications:

Simplify expressions to CoshIntegral:

Function Representations  (4)

Representation in terms of CosIntegral and Log:

CoshIntegral can be represented in terms of MeijerG:

CoshIntegral can be represented as a DifferentialRoot:

TraditionalForm formatting:

Applications  (3)

Plot the imaginary part in the complex plane:

Solve a differential equation:

Find the antiderivative using DSolveValue:

Compare with the answer given by Integrate:

Properties & Relations  (3)

Use FullSimplify to simplify expressions containing the hyperbolic cosine integral:

Use FunctionExpand to express CoshIntegral through other functions:

Find a numerical root:

Obtain CoshIntegral from integrals and sums:

Possible Issues  (2)

CoshIntegral can take large values for moderatesize arguments:

A larger setting for $MaxExtraPrecision can be needed:

Neat Examples  (2)

Nested integrals:

Plot the logarithm of the absolute value in the complex plane:

Wolfram Research (1996), CoshIntegral, Wolfram Language function, (updated 2022).


Wolfram Research (1996), CoshIntegral, Wolfram Language function, (updated 2022).


Wolfram Language. 1996. "CoshIntegral." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022.


Wolfram Language. (1996). CoshIntegral. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_coshintegral, author="Wolfram Research", title="{CoshIntegral}", year="2022", howpublished="\url{}", note=[Accessed: 23-May-2024 ]}


@online{reference.wolfram_2024_coshintegral, organization={Wolfram Research}, title={CoshIntegral}, year={2022}, url={}, note=[Accessed: 23-May-2024 ]}