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Repeated integration by parts generates successive inverse powers of a large endpoint, with a remainder expressed by the same integral with shifted parameters.

The Bernoulli numbers are a sequence of rational numbers that turns integer power sums into exact polynomials. The same numbers describe corrections when a discrete summation is compared with an integral. For example, the sum of squares $1^2+\cdots +N^2$ is $N^3/3+N^2/2+N/6$: the leading term is an integral, while the last two terms repair the difference between area and sampled values.

We use $B_1=-\frac{1}{2}$ throughout. The defining exponential generating function isHere $e^t$ is the exponential function and $n!$ is a factorial; in particular, the coefficient of $t^2$ is $B_2/2!$, rather than $B_2$. At $t=0$ the quotient has limiting value one. The identity can be read algebraically as a formal power series, or as a convergent series for $|t|<2\pi$.^[ref]

To compute the numbers, multiply by $e^t-1$ and compare coefficients. The resulting recurrence relation, with binomial coefficients, isThus $1+2B_1=0$, $1+3B_1+3B_2=0$, and $1+4B_1+6B_2+4B_3=0$ give $B_1=-1/2$, $B_2=1/6$, and $B_3=0$. Continuing givesThe recurrence relation agrees with equation 24.5.3 after replacing its index $n$ by $m+1$.^[ref]

Why do so many values vanish? Put $F(t)=t/(e^t-1)$. Direct substitution gives $F(-t)=F(t)+t$, so $F(t)+t/2$ is an even function. Its odd power series coefficients vanish: $B_{2r+1}=0$ for every positive integer $r$. The exception $B_1$ matters. The alternative convention obtained from $t/(1-e^{-t})=F(t)+t$ has $B_1^{+}=+1/2$ and exactly the same other values. A formula copied between conventions must also retain its summation endpoints.

Historically, integer power sums were studied before the modern Bernoulli number notation. A historical study examines formulas published in 1631 and distinguishes those separate formulas from encoding all exponents using one sequence of constants. This is a useful motivation for the generating function, without assigning all discovery to a single moment. ^[ref]

The Bernoulli polynomials extend the Bernoulli numbers by inserting the exponential function $e^{xt}$ into their exponential generating function:For example, $B_0(x)=1$, $B_1(x)=x-1/2$, $B_2(x)=x^2-x+1/6$, and $B_3(x)=x^3-3x^2/2+x/2$.^[ref]

Their crucial property is a finite difference identity:Indeed, subtracting the two exponential generating functions cancels $e^t-1$ and leaves $t{e}^{xt}$. Thus $B_{p+1}(x)/(p+1)$ plays for discrete summation the role that $x^{p+1}/(p+1)$ plays for integration: taking a unit finite difference recovers $x^p$.^[ref]

The Bessel function of the first kind $J_{\nu }$ is the solution of the Bessel differential equation selected by its regular power-series behavior at the origin. For $\mathrm{Re}\nu >-1/2$, it has the Poisson integral representation

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For $0<k<1$, the complete elliptic integral of the first kind is

It is the complete first-kind member of the elliptic integral family.

When $y=\lambda x$ with $\lambda >1$, the integral is dominated by its lower endpoint. Setting $t=y+s$ and applying Watson's lemma gives coefficients rational in $\lambda -1$.

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The fundamental strip of a Mellin transform is the interior of the set of real parts $\sigma$ for whichViewed in the complex plane, it is a vertical strip $\alpha <\mathrm{Re}s<\beta$; either bound may be infinite, and the interior may be empty. Both endpoints of the improper integral must be checked. Convergence on a boundary line, whether absolute convergence or conditional convergence, requires a separate test. The Mellin transform is holomorphic throughout a nonempty fundamental strip.^[ref]

A practical sufficient test uses Big O notation: if $f(x)=O(x^p)$ near zero and $f(x)=O(x^{-q})$ near infinity, with real $p,q$, then comparison with elementary improper integrals givesThus $-p<\sigma <q$ guarantees absolute convergence. These upper bounds on $|f|$ need not locate the maximal Mellin fundamental strip; faster actual decay can widen it. The strip property itself follows by bounding the weight at any intermediate $\sigma$ by the sum of two weights already known to be integrable. On a closed narrower strip, neighboring integrable weights also dominate the natural logarithm factors introduced by differentiation under the integral sign, making the transform a holomorphic function.

An exact two-endpoint example isHere both bounds are sharp: at $\sigma =-2$ the zero-end absolute convergence test contains ${\int }_0^{1}dx/x$, and at $\sigma =3$ the infinity-end test contains ${\int }_1^{\infty }dx/x$. The displayed rational function continues meromorphically outside the Mellin fundamental strip and has simple poles at $-2$ and $3$. Its other finite continued values outside the strip are values of a meromorphic continuation, not of a convergent defining integral. More general endpoint information leads to meromorphic continuation of a Mellin transform from an asymptotic expansion.

It follows by making the logarithm of the quotient entire and periodic; growth forces its periodic remainder to be constant.

The Gauss equation has singularities at $0,1,\infty$ and normalized exponent-zero solution $F(A,B;C;u)$ at the origin.

For a Dirichlet character $\chi$ of conductor $f$, define the generalized Bernoulli numbers byThey express special values of Dirichlet L-functions at nonpositive integers and occur in the interpolation formula for Kubota-Leopoldt p-adic L-functions.

For fixed order, the standard Hankel function expansions in $e^{\pm iz}{z}^{-1/2}$ are valid in different sectors on a chosen argument sheet. Analytic continuation can mix exponentials; the sectors must be specified rather than treating a formal inverse-power series as global.

A Hankel contour runs along both banks of a branch cut and circles its branch point, converting the jump of a complex power into a sine factor.

The two Hankel functions $H_{\nu }^{(1)}=J_{\nu }+i{Y}_{\nu }$ and $H_{\nu }^{(2)}=J_{\nu }-i{Y}_{\nu }$ solve the Bessel differential equation. Their large-argument forms distinguish outgoing and incoming oscillations and have sector-dependent asymptotic expansions.

Named after Charles Hermite.