# ChebyshevT

ChebyshevT[n,x]

gives the Chebyshev polynomial of the first kind .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• Explicit polynomials are given for integer n.
• .
• For certain special arguments, ChebyshevT automatically evaluates to exact values.
• ChebyshevT can be evaluated to arbitrary numerical precision.
• ChebyshevT automatically threads over lists.
• ChebyshevT[n,z] has a branch cut discontinuity in the complex z plane running from to if n is not an integer.
• ChebyshevT can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(7)

Evaluate numerically:

Compute the Chebyshev 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:

ChebyshevT can be used with Interval and CenteredInterval objects:

### Specific Values(7)

Values of ChebyshevT at fixed points:

ChebyshevT for symbolic n:

Values at zero:

Values at infinity:

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

Compute the associated ChebyshevT[7,x] polynomial:

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

### Visualization(3)

Plot the ChebyshevT 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)

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

ChebyshevT is defined for all complex values:

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:

ChebyshevT threads elementwise over lists:

Chebyshev polynomials are analytic:

In general, ChebyshevT 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 ChebyshevT 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)

ChebyshevT is defined through the identity:

The ordinary generating function of ChebyshevT:

The exponential generating function of ChebyshevT:

Recurrence relations:

## Generalizations & Extensions(2)

ChebyshevT can be applied to power series:

ChebyshevT can be applied to Interval:

## Applications(4)

Plot the first 10 Chebyshev polynomials:

Find a minimax approximation to the function Clip[4 x]:

Get an expansion for a function in the Chebyshev polynomials:

The values of the function at the Chebyshev nodes:

Find the Chebyshev coefficients:

Show the error:

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

## Properties & Relations(7)

Use FullSimplify with ChebyshevT:

Derivative of ChebyshevT is expressed in terms of ChebyshevU:

ChebyshevT can be represented as a DifferenceRoot:

General term in the series expansion of ChebyshevT:

The generating function for ChebyshevT:

The exponential generating function for ChebyshevT:

## Possible Issues(1)

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

Evaluate the function directly:

## Neat Examples(1)

Plot the first few BanchoffChmutov surfaces:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_chebyshevt, author="Wolfram Research", title="{ChebyshevT}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ChebyshevT.html}", note=[Accessed: 12-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_chebyshevt, organization={Wolfram Research}, title={ChebyshevT}, year={2022}, url={https://reference.wolfram.com/language/ref/ChebyshevT.html}, note=[Accessed: 12-July-2024 ]}