Convergent Integrals

Although this website focuses on divergent mathematical objects (such as divergent series, divergent products and divergent integrals), this page lists several convergent integrals that serve as input or intermediate calculation step to the determination of divergent series.

$\Gamma(z)$ denotes the Gamma function.
$\Gamma'(z)$ denotes the derivative of the Gamma function.
$\ln\Gamma(z)$ denotes the log-gamma function.
$\Psi^{(0)}(z) = \Psi(z) = \frac{d}{dz} \ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}$ denotes the digamma function.
$\Psi^{(1)}(z) = \frac{d}{dz} \Psi^{(0)}(z)$ denotes the trigamma function.
$\zeta(z)$ denotes the Riemann zeta function.
$\zeta'(z)$ denotes the derivative of the Riemann zeta function.
$\zeta(z,a)$ denotes the Hurwitz zeta function.
$\zeta'(z,a)$ denotes the derivative of the Hurwitz zeta function.
$\ln Z(z)$ denotes the log-zeta function.
$Z^{(0)}(z) = Z(z) = \frac{d}{dz} \ln\zeta(z) = \frac{\zeta'(z)}{\zeta(z)}$ denotes the dizeta function.
$P(z)$ denotes the prime zeta function.
$P'(z)$ denotes the derivative of the prime zeta function.
$\xi(z)$ denotes the Riemann xi function.
$\xi'(z)$ denotes the derivative of the Riemann xi function.
$\eta(z)$ denotes the Dirichlet eta function.
$\eta'(z)$ denotes the derivative of the Dirichlet eta function.
$K(z)$ denotes the K-function.
$K'(z)$ denotes the derivative of the K-function.
$\text{Ei}(z)$ denotes the exponential integral function.
$\text{Li}_s(z)$ denotes the polylogarithm function (also known as Jonquière’s function).
$\theta_3(z,q)$ denotes the Jacobi theta_3 function.
$\theta_3'(z,q)$ denotes the derivative of the Jacobi theta_3 function.

$\zeta(1) = \gamma \approx 0.57721566490153286060651209008240243\ldots$ denotes the Euler-Mascheroni constant.
$\zeta(2) = \dfrac{\pi^2}{6} \approx 1.6449340668482264364724151666460251\ldots$ denotes the Basel constant.
$\zeta(3) \approx 1.202056903159594285399738161511449990764\ldots$ denotes the Apéry constant.
$\varpi \approx 2.62205755429211981046483958989111941368275\ldots$ denotes the lemniscate constant.
$A = A_1 \approx 1.282427129100622636875342568869791727\ldots$ denotes the Glaisher-Kinkelin constant.
$C = \beta(2) \approx 0.91596559417721901505460351493238411\ldots$ denotes the Catalan constant.
$G \approx 0.59634736232319407434107849936927937607417\ldots$ denotes the Euler-Gompertz constant.

$A_n \; (n \geq 0)$ denotes the Glaisher-Kinkelin constants (with $A_0 = \sqrt{2\pi} \;$ and $A_1 = A$ ).
$B_n \; (n \geq 0)$ denotes the Bernoulli numbers (with $B_0 = 0$ and $B_1 = \pm \frac{1}{2}$ ).
$H_n \; (n \geq 0)$ denotes the harmonic numbers (with $H_0 = 0$ ).
$H_{n,r} = \displaystyle \sum_{m=1}^{n} \dfrac{1}{m^r}$ denotes the n-th generalized harmonic number or order r (with $H_{0,r} = 0$ ).
$H_n^{(r)}$ denotes the n-th hyperharmonic number of order r (with $H_0^{(r)} = 0$ ).
$M_n \; (n \geq 0)$ denotes the Meissel-Mertens constants (with $M_0 = \frac{1}{4} + \gamma$ ).
$P_n \; (n \geq 0)$ denotes the prime zeta constants (with $P_0 = -\frac{1}{2}$ ).
$\gamma_n \; (n \geq 0)$ denotes the Stieltjes constants (with $\gamma_0 = \gamma$ ).
$\lambda_n \; (n \geq 0)$ denotes the Li constants (with $\lambda_0 = 0$ ).

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