Assumptions:
TeX:
E_{2 k}\!\left(\tau\right) = 1 - \frac{4 k}{B_{2 k}} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} {n}^{2 k - 1} {q}^{m n}\; \text{ where } q = {e}^{2 \pi i \tau} k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Definitions:
Fungrim symbol | Notation | Short description |
---|---|---|
EisensteinE | Normalized Eisenstein series | |
BernoulliB | Bernoulli number | |
Sum | Sum | |
Pow | Power | |
Infinity | Positive infinity | |
Exp | Exponential function | |
Pi | The constant pi (3.14...) | |
ConstI | Imaginary unit | |
ZZGreaterEqual | Integers greater than or equal to n | |
HH | Upper complex half-plane |
Source code for this entry:
Entry(ID("15b347"), Formula(Equal(EisensteinE(Mul(2, k), tau), Where(Sub(1, Mul(Div(Mul(4, k), BernoulliB(Mul(2, k))), Sum(Sum(Mul(Pow(n, Sub(Mul(2, k), 1)), Pow(q, Mul(m, n))), For(m, 1, Infinity)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))), Variables(k, tau), Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))