differentiation formulas

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11: 3.4 Differentiation

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§3.4(i) Equally-Spaced Nodes

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12: 18.17 Integrals

… ►Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28). … ►Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in (Erdélyi et al., 1953b, §10.9(15)). …

13: 1.8 Fourier Series

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Parseval’s Formula

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§1.8(iii) Integration and Differentiation

… ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. ►

§1.8(iv) Poisson’s Summation Formula

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1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

Encodings:

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See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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14: 10.17 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for other ranges of $\mathrm{ph}z$ can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4). …

15: Philip J. Davis

… ►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. …Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky. …

16: 18.19 Hahn Class: Definitions

… ►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. … ►

Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

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17: 3.3 Interpolation

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Three-Point Formula

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Four-Point Formula

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Five-Point Formula

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Six-Point Formula

… ►For theory and applications see Stenger (1993, Chapter 3).

18: 27.14 Unrestricted Partitions

… ►Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula …Logarithmic differentiation of the generating function $1/f\left(x\right)$ leads to another recursion: … ►

§27.14(iii) Asymptotic Formulas

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19: 10.57 Uniform Asymptotic Expansions for Large Order

… ►Subsequently, for ${𝗂}_n^{(2)}\left((n+\frac{1}{2})z\right)$ the connection formula (10.47.11) is available. …

20: 9.12 Scorer Functions

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§9.12(v) Connection Formulas

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9.12.16 $${\mathrm{Gi}}^{\prime }\left(z\right)=\frac{3^{-1/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\cos \left(\frac{2k+1}{3}\pi \right)\mathrm{\Gamma }\left(\frac{k+2}{3}\right)\frac{{(3^{1/3}z)}^k}{k!}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Differentiate (9.12.15).

Permalink:

http://dlmf.nist.gov/9.12.E16

Encodings:

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See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.18 $${\mathrm{Hi}}^{\prime }\left(z\right)=\frac{3^{-1/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+2}{3}\right)\frac{{(3^{1/3}z)}^k}{k!}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Differentiate (9.12.17).

Permalink:

http://dlmf.nist.gov/9.12.E18

Encodings:

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See also:

Annotations for §9.12(vi), §9.12 and Ch.9

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

Encodings:

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See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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