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Fungrim entry: 2c26a1

11Tn ⁣(x)Tm ⁣(x)11x2dx={0,nmπ,n=m=0π2,n=mandn0\int_{-1}^{1} T_{n}\!\left(x\right) T_{m}\!\left(x\right) \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = \begin{cases} 0, & n \ne m\\\pi, & n = m = 0\\\frac{\pi}{2}, & n = m \,\mathbin{\operatorname{and}}\, n \ne 0\\ \end{cases}
Assumptions:nZ0andmZ0n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
TeX:
\int_{-1}^{1} T_{n}\!\left(x\right) T_{m}\!\left(x\right) \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = \begin{cases} 0, & n \ne m\\\pi, & n = m = 0\\\frac{\pi}{2}, & n = m \,\mathbin{\operatorname{and}}\, n \ne 0\\ \end{cases}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, dx Integral
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("2c26a1"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), ChebyshevT(m, x)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), Tuple(x, -1, 1)), Cases(Tuple(0, Unequal(n, m)), Tuple(ConstPi, Equal(n, m, 0)), Tuple(Div(ConstPi, 2), And(Equal(n, m), Unequal(n, 0)))))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

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2019-08-19 14:38:23.809000 UTC