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 .

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 (4)

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:

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 (4)

Plot the ChebyshevT function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of ChebyshevT function have different branch cut structures:

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:

TraditionalForm formatting:

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 (3)

Plot the first 10 Chebyshev polynomials:

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

Solve a differential equation with the ChebyshevT function as 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:

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ChebyshevT

Details

Examples

Basic Examples (7)