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Fungrim entry: 3d77ab

11Tn ⁣(x)xm11x2dx=0\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0
Assumptions:nZ0  and  m{0,1,,n1}n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}
\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
    Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), Pow(x, m)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), For(x, -1, 1)), 0)),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, Range(0, Sub(n, 1))))))

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2021-03-15 19:12:00.328586 UTC