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 10^(th) 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  (44)

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  (4)

Plot the ChebyshevU 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 ChebyshevU function have different branch cut structures:

Function Properties  (14)

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

ChebyshevU is defined for all complex values besides :

TemplateBox[{1, x}, ChebyshevU] achieves all real and complex values:

Real range of TemplateBox[{2, x}, ChebyshevU]:

It achieves all complex values:

Chebyshev polynomial of an odd order is odd:

Chebyshev polynomial of an even order is even:

ChebyshevU threads elementwise over lists:

Chebyshev polynomials are analytic:

In general, ChebyshevU is neither analytic nor meromorphic:

TemplateBox[{2, x}, ChebyshevU] is neither non-decreasing nor non-increasing:

TemplateBox[{2, x}, ChebyshevU] is not injective:

TemplateBox[{1, x}, ChebyshevU] is:

TemplateBox[{2, x}, ChebyshevU] is not surjective:

TemplateBox[{1, x}, ChebyshevU] is:

TemplateBox[{2, x}, ChebyshevU] is neither non-negative nor non-positive:

TemplateBox[{n, x}, ChebyshevU] has singularities and discontinuities for when is not an integer:

TemplateBox[{2, x}, ChebyshevU] 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 ^(th) 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 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  (4)

Approximate a function on the interval :

Build a curve that passes through given points:

Light amplitude transmission through layers of glass:

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

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_2022_chebyshevu, author="Wolfram Research", title="{ChebyshevU}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ChebyshevU.html}", note=[Accessed: 19-August-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_chebyshevu, organization={Wolfram Research}, title={ChebyshevU}, year={2022}, url={https://reference.wolfram.com/language/ref/ChebyshevU.html}, note=[Accessed: 19-August-2022 ]}