gives the hyperbolic sine integral function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • SinhIntegral[z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, SinhIntegral automatically evaluates to exact values.
  • SinhIntegral can be evaluated to arbitrary numerical precision.
  • SinhIntegral automatically threads over lists.


open allclose all

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 Infinity:

Scope  (39)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

SinhIntegral can take complex number inputs:

Evaluate SinhIntegral efficiently at high precision:

SinhIntegral threads elementwise over lists:

Specific Values  (3)

Value at the origin:

Values at infinity:

Find the real root of the equation TemplateBox[{x}, SinhIntegral]=0.8:

Visualization  (2)

Plot the SinhIntegral function:

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}}, SinhIntegral]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, SinhIntegral]:

Function Properties  (10)

SinhIntegral is defined for all real and complex values:

SinhIntegral takes all the real values:

SinhIntegral is an odd function:

SinhIntegral is an analytic function of x:

SinhIntegral is non-decreasing:

SinhIntegral is injective:

SinhIntegral is surjective:

SinhIntegral is neither non-negative nor non-positive:

SinhIntegral has no singularities or discontinuities:

SinhIntegral is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of SinhIntegral:

Definite integral of an odd integrand over an interval centered at the origin is 0:

More integrals:

Series Expansions  (4)

Taylor expansion for SinhIntegral:

Plot the first three approximations for SinhIntegral around :

General term in the series expansion of SinhIntegral:

Find series expansions at infinity:

Give the result for an arbitrary symbolic direction :

SinhIntegral can be applied to power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:


Function Identities and Simplifications  (3)

Primary definition of SinhIntegral:

Argument simplifications:

Simplify expressions to SinhIntegral:

Function Representations  (5)

Representation in terms of SinIntegral:

Series representation of SinhIntegral:

SinhIntegral can be represented in terms of MeijerG:

SinhIntegral can be represented as a DifferentialRoot:

TraditionalForm formatting:

Applications  (2)

Plot the real part in the complex plane:

Solve a differential equation:

Properties & Relations  (5)

Use FullSimplify to simplify expressions containing hyperbolic sine integrals:

Find a numerical root:

Obtain SinhIntegral from integrals and sums:


Laplace transform:

Possible Issues  (3)

SinhIntegral can take large values for moderatesize arguments:

A larger setting for $MaxExtraPrecision can be needed:

In traditional form, parentheses are required:

Neat Examples  (1)

Nested integrals:

Wolfram Research (1996), SinhIntegral, Wolfram Language function,


Wolfram Research (1996), SinhIntegral, Wolfram Language function,


@misc{reference.wolfram_2021_sinhintegral, author="Wolfram Research", title="{SinhIntegral}", year="1996", howpublished="\url{}", note=[Accessed: 28-September-2021 ]}


@online{reference.wolfram_2021_sinhintegral, organization={Wolfram Research}, title={SinhIntegral}, year={1996}, url={}, note=[Accessed: 28-September-2021 ]}


Wolfram Language. 1996. "SinhIntegral." Wolfram Language & System Documentation Center. Wolfram Research.


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