Leibniz formula

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1: 1.4 Calculus of One Variable

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Leibniz’s Formula

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Faà Di Bruno’s Formula

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2: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for . … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

3: 1.2 Elementary Algebra

… ►where $det(𝐀)$ is defined by the Leibniz formula … ►Formula (1.2.77) is more generally valid for all square matrices $𝐀$, not necessarily non-defective, see Hall (2015, Thm 2.12).

4: Howard S. Cohl

… ►Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.

5: Preface

… ►Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. …

6: 5.5 Functional Relations

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§5.5(ii) Reflection

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

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See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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§5.5(iii) Multiplication

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Duplication Formula

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Gauss’s Multiplication Formula

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7: 24.6 Explicit Formulas

§24.6 Explicit Formulas

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

Encodings:

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See also:

Annotations for §24.6 and Ch.24

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24.6.7 $$B_n\left(x\right)=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(x+j)}^n,$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E7

Encodings:

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See also:

Annotations for §24.6 and Ch.24

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24.6.12 $$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

8: 27.5 Inversion Formulas

§27.5 Inversion Formulas

… ►which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: …

9: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …

10: 18.42 Software

… ►A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …