# ChebyshevU

ChebyshevU[n,x]

gives the Chebyshev polynomial of the second kind .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• Explicit polynomials are given for integer n.
• .
• For certain special arguments, ChebyshevU automatically evaluates to exact values.
• ChebyshevU can be evaluated to arbitrary numerical precision.
• ChebyshevU automatically threads over lists.
• ChebyshevU[n,z] has a branch cut discontinuity in the complex z plane running from to for noninteger n.
• ChebyshevU can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(7)

Evaluate numerically:

Compute the ChebyshevU polynomial:

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(43)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

ChebyshevU can be used with Interval and CenteredInterval objects:

### Specific Values(7)

Values of ChebyshevU at fixed points:

ChebyshevU for symbolic n:

Values at zero:

Values at infinity:

Find the first positive maximum of ChebyshevU[5,x]:

Compute the associated ChebyshevU[7,x] polynomial:

Compute the associated ChebyshevU[1/2,x] polynomial for half-integer n:

### Visualization(3)

Plot the ChebyshevU function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot the Chebyshev polynomial as a function of two variables:

### Function Properties(14)

ChebyshevU is defined for all real values from the interval [-1,]:

ChebyshevU is defined for all complex values besides :

achieves all real and complex values:

Real range of :

It achieves all complex values:

Chebyshev polynomial of an odd order is odd:

Chebyshev polynomial of an even order is even:

Chebyshev polynomials are analytic:

In general, ChebyshevU is neither analytic nor meromorphic:

is neither non-decreasing nor non-increasing:

is not injective:

is:

is not surjective:

is:

is neither non-negative nor non-positive:

has singularities and discontinuities for when is not an integer:

is convex:

### Differentiation(3)

First derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x when n=5:

Formula for the derivative with respect to x:

### Integration(4)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Definite integral of ChebyshevU over a period for odd integers is 0:

More integrals:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Taylor expansion at a generic point:

### Function Identities and Simplifications(4)

ChebyshevU is defined through the following trigonometric identity:

The ordinary generating function of ChebyshevU:

The exponential generating function of ChebyshevU:

Recurrence relations:

## Generalizations & Extensions(2)

ChebyshevU can be applied to power series:

ChebyshevU can be applied to Interval:

## Applications(7)

Approximate a function on the interval :

Build a curve that passes through given points:

Light amplitude transmission through layers of glass:

Define a Toeplitz tridiagonal matrix:

Show the 4×4 case:

The characteristic polynomial of a Toeplitz tridiagonal matrix can be expressed in terms of ChebyshevU:

Verify for the first few cases:

Define the KacMurdockSzegő (KMS) matrix, a symmetric Toeplitz matrix:

The KMS matrix is the correlation matrix of an autoregressive process of order one (i.e. an AR(1) process):

The characteristic polynomial of the KMS matrix can be expressed in terms of ChebyshevU:

Solve a differential equation with the ChebyshevU function as the inhomogeneous part:

Find a Chebyshev polynomial from its generating function:

## Properties & Relations(7)

Get the list of coefficients in a ChebyshevU polynomial:

Use FunctionExpand to expand through trigonometric functions:

Derivative of ChebyshevU with respect to :

ChebyshevU can be represented as a DifferenceRoot:

General term in the series expansion of ChebyshevU:

The generating function for ChebyshevU:

The exponential generating function for ChebyshevU:

## Possible Issues(1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Wolfram Research (1988), ChebyshevU, Wolfram Language function, https://reference.wolfram.com/language/ref/ChebyshevU.html (updated 2022).

#### Text

Wolfram Research (1988), ChebyshevU, Wolfram Language function, https://reference.wolfram.com/language/ref/ChebyshevU.html (updated 2022).

#### CMS

Wolfram Language. 1988. "ChebyshevU." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ChebyshevU.html.

#### APA

Wolfram Language. (1988). ChebyshevU. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ChebyshevU.html

#### BibTeX

@misc{reference.wolfram_2024_chebyshevu, author="Wolfram Research", title="{ChebyshevU}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ChebyshevU.html}", note=[Accessed: 17-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_chebyshevu, organization={Wolfram Research}, title={ChebyshevU}, year={2022}, url={https://reference.wolfram.com/language/ref/ChebyshevU.html}, note=[Accessed: 17-July-2024 ]}