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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\!\left(k\right) = \int_{N}^{U} f\!\left(t\right) \, dt + \frac{f\!\left(N\right) + f\!\left(U\right)}{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:NZandUZandNUandMZ1and[N,U]HolomorphicDomain ⁣(f ⁣(t),t,C)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}}\, \left[N, U\right] \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(t\right), t, \mathbb{C}\right)
TeX:
\sum_{k=N}^{U} f\!\left(k\right) = \int_{N}^{U} f\!\left(t\right) \, dt + \frac{f\!\left(N\right) + f\!\left(U\right)}{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}}\, \left[N, U\right] \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(t\right), t, \mathbb{C}\right)
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, dx Integral
BernoulliBBnB_{n} Bernoulli number
Factorialn!n ! Factorial
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) 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
ClosedInterval[a,b]\left[a, b\right] Closed interval
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("ce2272"),
    Formula(Equal(Sum(f(k), Tuple(k, N, U)), Add(Add(Integral(f(t), Tuple(t, N, U)), Add(Div(Add(f(N), f(U)), 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Sub(Derivative(f(t), Tuple(t, U, Sub(Mul(2, k), 1))), Derivative(f(t), Tuple(t, N, Sub(Mul(2, k), 1))))), Tuple(k, 1, M)))), Integral(Mul(Div(BernoulliPolynomial(Mul(2, M), Sub(t, Floor(t))), Factorial(Mul(2, M))), Derivative(f(t), Tuple(t, t, Mul(2, M)))), Tuple(t, N, U))))),
    Variables(f, N, U, M),
    Assumptions(And(Element(N, ZZ), Element(U, ZZ), LessEqual(N, U), Element(M, ZZGreaterEqual(1)), Subset(ClosedInterval(N, U), HolomorphicDomain(f(t), t, CC)))))

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2019-08-21 11:44:15.926409 UTC