Fungrim entry: 0ed026

\left(1 - {x}^{2}\right) y''(x) - x y'(x) + {n}^{2} y(x) = 0\; \text{ where } y(x) = {c}_{1} T_{n}\!\left(x\right) + {c}_{2} U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right)

TeX:

\left(1 - {x}^{2}\right) y''(x) - x y'(x) + {n}^{2} y(x) = 0\; \text{ where } y(x) = {c}_{1} T_{n}\!\left(x\right) + {c}_{2} U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("0ed026"),
    Formula(Where(Equal(Add(Sub(Mul(Sub(1, Pow(x, 2)), ComplexDerivative(y(x), For(x, x, 2))), Mul(x, ComplexDerivative(y(x), For(x, x, 1)))), Mul(Pow(n, 2), y(x))), 0), Equal(y(x), Add(Mul(Subscript(c, 1), ChebyshevT(n, x)), Mul(Mul(Subscript(c, 2), ChebyshevU(Sub(n, 1), x)), Sqrt(Sub(1, Pow(x, 2)))))))),
    Variables(n, x, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(Element(n, ZZ), Element(x, CC), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), Or(Equal(Subscript(c, 2), 0), NotElement(x, Union(OpenClosedInterval(Neg(Infinity), 1), ClosedOpenInterval(1, Infinity)))))))

Topics using this entry

Chebyshev polynomials