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1: 27.13 Functions

… ►Jacobi (1829) notes that $r_2\left(n\right)$ is the coefficient of $x^n$ in the square of the theta function $\vartheta \left(x\right)$: …(In §20.2(i), $\vartheta \left(x\right)$ is denoted by ${\theta }_3(0,x)$.) … ►

27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

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See also:

Annotations for §27.13(iv), §27.13 and Ch.27

… ►

27.13.6 $${(\vartheta \left(x\right))}^2=1+4\sum _{n=1}^{\mathrm{\infty }}\left({\delta }_1(n)-{\delta }_3(n)\right)x^n,$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $n$: positive integer, $x$: real number and ${\delta }_j(n)$: number of divisors

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E6

Encodings:

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See also:

Annotations for §27.13(iv), §27.13 and Ch.27

… ►Mordell (1917) notes that $r_k\left(n\right)$ is the coefficient of $x^n$ in the power-series expansion of the $k$th power of the series for $\vartheta \left(x\right)$. …

2: 4.42 Solution of Triangles

… ►

4.42.1 $$\sin A=\frac{a}{c}=\frac{1}{\csc A},$$

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Symbols:

$\csc z$: cosecant function, $\sin z$: sine function, $A$: angle, $a$: height and $c$: hypotenuse

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E1

Encodings:

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See also:

Annotations for §4.42(i), §4.42 and Ch.4

►

4.42.2 $$\cos A=\frac{b}{c}=\frac{1}{\sec A},$$

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Symbols:

$\cos z$: cosine function, $\sec z$: secant function, $A$: angle, $b$: base and $c$: hypotenuse

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E2

Encodings:

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See also:

Annotations for §4.42(i), §4.42 and Ch.4

►

4.42.3 $$\tan A=\frac{a}{b}=\frac{1}{\cot A}.$$

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Symbols:

$\cot z$: cotangent function, $\tan z$: tangent function, $A$: angle, $a$: height and $b$: base

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E3

Encodings:

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See also:

Annotations for §4.42(i), §4.42 and Ch.4

… ►

4.42.5 $$c^2=a^2+b^2-2ab\cos C,$$

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Symbols:

$\cos z$: cosine function, $a$: length, $b$: base, $c$: length and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E5

Encodings:

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See also:

Annotations for §4.42(ii), §4.42 and Ch.4

►

4.42.6 $$a=b\cos C+c\cos B$$

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Symbols:

$\cos z$: cosine function, $a$: length, $b$: base, $c$: length, $B$: angle and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E6

Encodings:

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See also:

Annotations for §4.42(ii), §4.42 and Ch.4

…

3: 15 Hypergeometric Function

Chapter 15 Hypergeometric Function

4: 13 Confluent Hypergeometric Functions

Chapter 13 Confluent Hypergeometric Functions

5: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►

10.46.3 $$E_{a,b}\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(ak+b\right)},$$ $a>0$.

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Defines:

$E_{a,b}\left(z\right)$: Mittag-Leffler function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.46.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.46 and Ch.10

… ►This reference includes exponentially-improved asymptotic expansions for $E_{a,b}\left(z\right)$ when $|z|\to \mathrm{\infty }$, together with a smooth interpretation of Stokes phenomena. …

6: 8.17 Incomplete Beta Functions

… ►where, as in §5.12, $\mathrm{B}(a,b)$ denotes the beta function: … ►

8.17.4 $$I_x(a,b)=1-I_{1-x}(b,a).$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.3 (Symmetry relation)

Referenced by:

§8.17(v), §8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E4

Encodings:

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Annotations for §8.17(i), §8.17 and Ch.8

… ►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). … ►

8.17.13 $$(a+b)I_x(a,b)=a{I}_x(a+1,b)+b{I}_x(a,b+1),$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.7

Permalink:

http://dlmf.nist.gov/8.17.E13

Encodings:

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See also:

Annotations for §8.17(iv), §8.17 and Ch.8

… ►

8.17.16 $$a{I}_x(a+1,b)=(a+cx)I_x(a,b)-cx{I}_x(a-1,b),$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter, $x$: variable and $c$

Permalink:

http://dlmf.nist.gov/8.17.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(iv), §8.17 and Ch.8

…

7: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). … ►To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … ►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). …

8: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …

9: 21.8 Abelian Functions

… ►An Abelian function is a $2g$-fold periodic, meromorphic function of $g$ complex variables. …For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

10: Mourad E. H. Ismail

… ► Garvan), Kluwer Academic Publishers, 2001; and Theory and Applications of Special Functions: A volume dedicated to Mizan Rahman (with E. …