Fungrim entry: 30b67b

$\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y(x) = 0\; \text{ where } y(x) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{n + 1}\!\left(x\right)}{\sqrt{1 - {x}^{2}}}$
Assumptions:$n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left({c}_{2} = 0 \,\mathbin{\operatorname{or}}\, x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right) \,\mathbin{\operatorname{and}}\, x \notin \left\{-1, 1\right\}$
TeX:
\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y(x) = 0\; \text{ where } y(x) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{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) \,\mathbin{\operatorname{and}}\, x \notin \left\{-1, 1\right\}
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
ChebyshevU$U_{n}\!\left(x\right)$ Chebyshev polynomial of the second kind
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first 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("30b67b"),
Formula(Where(Equal(Add(Sub(Mul(Sub(1, Pow(x, 2)), ComplexDerivative(y(x), For(x, x, 2))), Mul(Mul(3, x), ComplexDerivative(y(x), For(x, x, 1)))), Mul(Mul(n, Add(n, 2)), y(x))), 0), Equal(y(x), Add(Mul(Subscript(c, 1), ChebyshevU(n, x)), Mul(Subscript(c, 2), Div(ChebyshevT(Add(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)))), NotElement(x, Set(-1, 1)))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-12-11 23:01:54.699850 UTC