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Fungrim entry: 0ed026

(1x2)y(x)xy(x)+n2y(x)=0   where y(x)=c1Tn ⁣(x)+c2Un1 ⁣(x)1x2\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:nZandxCandc1Candc2Cand(c2=0orx(,1][1,))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
Powab{a}^{b} Power
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sqrtz\sqrt{z} Principal square root
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\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)))))))

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

2019-11-19 15:10:20.037976 UTC