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The upper incomplete gamma function is $\gamma (x,y)={\int }_y^{\infty }{t}^{x-1}{e}^{-t}dt$ under the convention used here.

An integer power sum adds a fixed nonnegative integer power of consecutive integers. Let $N$ be a nonnegative integer and $p$ a nonnegative integer. For $p=0$, use the constant polynomial $x^0=1$, including at zero, so $S_0(N)=N$. Summing the Bernoulli polynomial finite difference identity makes the intermediate terms cancel and givesThe sum is empty when $N=0$. For the endpoints $1,\dots ,N$, the formula valid also at $p=0$ is insteadFor $p\ge 1$, $B_{p+1}(1)=B_{p+1}(0)$, and this becomes the standard positive-endpoint formula. The exception at $p=0$ is essential: $B_1(1)-B_1(0)=1$.^[ref]

For $p\ge 1$, adding $N^p$ to $S_p(N)$ changes the $B_1$ contribution from $-N^p/2$ to $+N^p/2$. Equivalently,where only $B_1^{+}$ differs from our Bernoulli numbers. A quick endpoint check is $p=1$: the sums over $0,\dots ,N-1$ and $1,\dots ,N$ are $N(N-1)/2$ and $N(N+1)/2$, respectively. The latter is a triangular number.

The Jacobi elliptic sine is the local inverse of the elliptic integral of the first kind: ifthen $z=\mathrm{sn}(u,k)$. Its analytic continuation is a meromorphic doubly periodic function.

It is the sn member of the Jacobi elliptic functions.

The Laplace contour-integral method seeks a solution in the formReplacing multiplication by $z$ with differentiation of $e^{zt}$ in $t$ and integrating by parts turns the differential equation for $w$ into a first-order equation for $f$. The contour $\gamma$ must make the resulting endpoint term vanish; finite endpoints are often branch points of $f$, while infinite ends must enter sectors where the exponential decays.

If a smooth real phase $\varphi$ has a unique nondegenerate interior maximum at $t_0$, thenFor a maximum at an endpoint $b$ with ${\varphi }^{\prime }(b)>0$, the leading factor is $a(b)e^{x\varphi (b)}/(x{\varphi }^{\prime }(b))$.

For fixed $x$ and $y\to \infty$,

Homepage: dlmf.nist.gov/about/bio/LCMaximon

The Mellin transform is an integral transform suited to changes of scale: replacing $f(x)$ by $f(ax)$ multiplies its transform by $a^{-s}$ for $a>0$. For a locally integrable function $f:(0,\infty )\to \mathbb{C}$, definewhere the improper integral converges. The complex number $x^{s-1}=\exp ((s-1)\log x)$ uses the real natural logarithm because $x>0$. Its absolute value is $x^{\sigma -1}$, so absolute convergence depends on the real part $\sigma$ rather than the imaginary part $\tau$. The Mellin fundamental strip records the open range of such real parts.^[ref]

A familiar example is the exponential function $f(x)=e^{-x}$:This is exactly the Gamma integral. Near zero $e^{-x}$ tends to one, so absolute convergence requires $\sigma >0$; at infinity $x^r{e}^{-x}\to 0$ for every real $r$, so there is no upper boundary. The Gamma function has a meromorphic continuation with simple poles at $0,-1,-2,\dots$, but a finite continued value elsewhere in $\sigma \le 0$ does not make the defining improper integral converge.^[ref]

The Mellin transform of an interval indicator gives an elementary example with absolute convergence for every complex number $s$. The Mellin transform in logarithmic coordinates explains why changes of scale become translations, connecting the scaling property of the Mellin transform to the Fourier transform.

The modified Bessel equation of order zero isIts solution regular at zero and normalized by $I_0(0)=1$ is the modified Bessel function $I_0$.

The modified Bessel function of the second kind is the solution $K_{\nu }(x)$ of the modified Bessel differential equation that decays exponentially as $x\to +\infty$.

The solutions of $\sinh w=z$ arefor any chosen local branch $\mathrm{Arcsinh}$.

It is the complex multivalued sine member of the inverse hyperbolic functions.

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Chapter authors: Tom H. Koornwinder, Roderick S. C. Wong, Roelof Koekoek, René F. Swarttouw, William P. Reinhardt. These are Chapter 18 credits in the edited NIST Digital Library of Mathematical Functions. The National Institute of Standards and Technology issues and hosts the reference. No chapter publication date is established; version 1.2.8, released 15 September 2026, is an online reference release.

Chapter and author credits: dlmf.nist.gov/18

Cited section: §18.2(i), Orthogonality on Intervals: equation 18.2.1; §18.2(i), equation 18.2.1_5 and surrounding weight/moment hypotheses; Chapter 18 current five-person author byline.

Cited section: Table 18.3.1, Legendre row: (-1,1), unit weight, squared norm 2/(2n+1), leading coefficient; §18.3 paragraph introducing orthogonality and standardization; Chapter 18 author byline.

Chapter 2 contributors: Frank W. J. Olver, Roderick S. C. Wong. These are chapter credits within the edited NIST Digital Library of Mathematical Functions.^[ref]

Publisher: National Institute of Standards and Technology.

Online text: dlmf.nist.gov/2.11

Online reference release inspected: version 1.2.8, 15 September 2026. This date identifies the site release.

Cited passages: NIST Digital Library of Mathematical Functions, §2.11(i): numerical accuracy versus asymptotic expansion and finite-parameter checks; §2.11(iii): exponentially improved expansions.

Chapter 2 contributors: Frank W. J. Olver, Roderick S. C. Wong. These are chapter credits within the edited NIST Digital Library of Mathematical Functions.^[ref]

Publisher: National Institute of Standards and Technology.

Online text: dlmf.nist.gov/2.1

Online reference release inspected: version 1.2.8, 15 September 2026. This date identifies the site release.

Cited passages: NIST Digital Library of Mathematical Functions, §2.1(iii), equations 2.1.13–2.1.15: fixed-order inverse-power expansions and uniqueness; §2.1(v): generalized asymptotic scales and expansions.

Section of NIST Digital Library of Mathematical Functions, Chapter 3, Numerical Methods, whose credited chapter author is Nico M. Temme (N. M. Temme on the chapter landing page). This credit concerns Chapter 3, not the whole edited collection. The National Institute of Standards and Technology issues and hosts the reference. No separate section publication or revision date is established.

Online text: dlmf.nist.gov/3.3

Chapter authorship: dlmf.nist.gov/3

Cited passages: NIST Digital Library of Mathematical Functions, Chapter 3, §3.3 Interpolation: Chapter 3 landing page: N. M. Temme byline and acknowledgements distinguish author from contributors; §3.3(iii), equation 3.3.34: distinct-node base cases and recurrence; §3.3(iii), equation 3.3.35: explicit symmetric sum; §3.3(iv), equation 3.3.38 and following new-node update paragraph.

Section of NIST Digital Library of Mathematical Functions. Chapter 3 author: Nico M. Temme, credited as N. M. Temme. National Institute of Standards and Technology issues and hosts the edited reference. The inspected online release is version 1.2.8, 15 September 2026; this is a site release date, not a section publication date. No separate subsection publication or revision date is established.

Online text: dlmf.nist.gov/3.5.ii

Chapter credit: dlmf.nist.gov/3

Full-name identity: dlmf.nist.gov/about/bio/NMTemme

Cited passages: NIST Digital Library of Mathematical Functions, §3.5(ii), equations 3.5.6–3.5.8.