CosIntegral

CosIntegral[z]

gives the cosine integral function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • CosIntegral[z] has a branch cut discontinuity in the complex z plane running from - to 0.
  • For certain special arguments, CosIntegral automatically evaluates to exact values.
  • CosIntegral can be evaluated to arbitrary numerical precision.
  • CosIntegral automatically threads over lists.

Examples

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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  (35)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate CosIntegral efficiently at high precision:

CosIntegral threads elementwise over lists and matrices:

Specific Values  (3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of (dTemplateBox[{x}, CosIntegral])/(dx)=0:

Visualization  (2)

Plot the CosIntegral function:

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

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

Function Properties  (8)

CosIntegral is defined for all positive real values:

Complex domain:

CosIntegral is not an analytic function:

Nor is it meromorphic:

CosIntegral is neither non-decreasing nor non-increasing:

CosIntegral is not injective:

CosIntegral is not surjective:

CosIntegral is neither non-negative nor non-positive:

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

CosIntegral is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of CosIntegral:

Definite integral of CosIntegral over its entire real domain:

More integrals:

Series Expansions  (3)

Taylor expansion for CosIntegral around :

Plots of the first three approximations for CosIntegral around :

Find series expansion at infinity:

CosIntegral can be applied to power series:

Function Identities and Simplifications  (4)

Use FullSimplify to simplify expressions containing the cosine integral:

Use FunctionExpand to express CosIntegral through other functions:

Simplify expressions to CosIntegral:

Argument simplifications:

Function Representations  (5)

Primary definition of CosIntegral:

Series representation of CosIntegral:

CosIntegral can be represented in terms of MeijerG:

CosIntegral can be represented as a DifferentialRoot:

TraditionalForm formatting:

Generalizations & Extensions  (1)

Find series expansions at infinity:

Applications  (4)

Average radiated power for a thin linear half-wave antenna:

Plot the imaginary part in the complex plane:

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

Solve a differential equation:

Properties & Relations  (7)

Use FullSimplify to simplify expressions containing the cosine integral:

Use FunctionExpand to express CosIntegral through other functions:

Find a numerical root:

Obtain CosIntegral from integrals and sums:

Obtain CosIntegral from a differential equation:

Calculate the Wronskian:

Laplace transform:

Possible Issues  (2)

CosIntegral can take large values for moderatesize arguments:

A larger setting for $MaxExtraPrecision can be needed:

Neat Examples  (1)

Nested integrals:

Wolfram Research (1991), CosIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/CosIntegral.html.

Text

Wolfram Research (1991), CosIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/CosIntegral.html.

BibTeX

@misc{reference.wolfram_2021_cosintegral, author="Wolfram Research", title="{CosIntegral}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/CosIntegral.html}", note=[Accessed: 13-June-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_cosintegral, organization={Wolfram Research}, title={CosIntegral}, year={1991}, url={https://reference.wolfram.com/language/ref/CosIntegral.html}, note=[Accessed: 13-June-2021 ]}

CMS

Wolfram Language. 1991. "CosIntegral." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CosIntegral.html.

APA

Wolfram Language. (1991). CosIntegral. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CosIntegral.html