CosIntegral

CosIntegral[z]

gives the cosine integral function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{z}, CosIntegral]=-int_z^inftycos(t)/tdt$ .
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.
CosIntegral can be used with Interval and CenteredInterval objects. »

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 (37)

Numerical Evaluation (6)

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix CosIntegral function using MatrixFunction:

Specific Values (3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of :

Visualization (2)

Plot the CosIntegral function:

Plot the real part of :

Plot the imaginary part of :

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 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 (6)

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:

Real part of the Euler–Heisenberg effective action:

Find the leading term in :

Plot Nielsen's spiral:

The curvature is a simple function of the parameter:

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 moderate‐size arguments:

A larger setting for $MaxExtraPrecision can be needed:

Neat Examples (1)

Nested integrals:

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