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Fungrim entry: ce2272

k=NUf(k)=NUf(t)dt+f(N)+f(U)2+k=1MB2k(2k)!(f(2k1)(U)f(2k1)(N))+NUB2M ⁣(tt)(2M)!f(2M)(t)dt\sum_{k=N}^{U} f(k) = \int_{N}^{U} f(t) \, dt + \frac{f(N) + f(U)}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \left({f}^{(2 k - 1)}(U) - {f}^{(2 k - 1)}(N)\right) + \int_{N}^{U} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} {f}^{(2 M)}(t) \, dt
Assumptions:NZandUZandNUandMZ1andf(t) is holomorphic on t[N,U]N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, U \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, N \le U \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, f(t) \text{ is holomorphic on } t \in \left[N, U\right]
TeX:
\sum_{k=N}^{U} f(k) = \int_{N}^{U} f(t) \, dt + \frac{f(N) + f(U)}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \left({f}^{(2 k - 1)}(U) - {f}^{(2 k - 1)}(N)\right) + \int_{N}^{U} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} {f}^{(2 M)}(t) \, dt

N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, U \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, N \le U \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, f(t) \text{ is holomorphic on } t \in \left[N, U\right]
Definitions:
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
BernoulliBBnB_{n} Bernoulli number
Factorialn!n ! Factorial
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
ZZZ\mathbb{Z} Integers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
ClosedInterval[a,b]\left[a, b\right] Closed interval
Source code for this entry:
Entry(ID("ce2272"),
    Formula(Equal(Sum(f(k), For(k, N, U)), Add(Add(Integral(f(t), For(t, N, U)), Add(Div(Add(f(N), f(U)), 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Sub(ComplexDerivative(f(t), For(t, U, Sub(Mul(2, k), 1))), ComplexDerivative(f(t), For(t, N, Sub(Mul(2, k), 1))))), For(k, 1, M)))), Integral(Mul(Div(BernoulliPolynomial(Mul(2, M), Sub(t, Floor(t))), Factorial(Mul(2, M))), ComplexDerivative(f(t), For(t, t, Mul(2, M)))), For(t, N, U))))),
    Variables(f, N, U, M),
    Assumptions(And(Element(N, ZZ), Element(U, ZZ), LessEqual(N, U), Element(M, ZZGreaterEqual(1)), IsHolomorphic(f(t), ForElement(t, Subset(ClosedInterval(N, U)))))))

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2019-10-05 13:11:19.856591 UTC