Fungrim entry: ce2272

\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:

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
`Sum`	$\sum_{n} f(n)$	Sum
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`ClosedInterval`	$\left[a, b\right]$	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)))))))

Topics using this entry

General analytic functions