functions%20s%28%CF%B5%2C%E2%84%93%3Br%29%2Cc%28%CF%B5%2C%E2%84%93%3Br%29

SearchAdvancedHelp

(0.024 seconds)

21—30 of 1006 matching pages

21: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►

§8.17(ii) Hypergeometric Representations

… ►

§8.17(iii) Integral Representation

… ►

§8.17(iv) Recurrence Relations

… ►

§8.17(vi) Sums

…

22: 12.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. …Whittaker’s notation $D_{\nu }\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).

23: 17.1 Special Notation

§17.1 Special Notation

… ► ►►

$k,j,m,n,r,s$	nonnegative integers.
…

►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. ►Another function notation used is the “idem” function: …

24: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►

Bessel Functions

… ►

Confluent Hypergeometric Functions

… ►

Positive $a$,

… ►

§12.14(x) Modulus and Phase Functions

…

25: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. … ►

Other Notations

►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and …

26: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation … ►

§11.10(vi) Relations to Other Functions

… ► … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

…

27: 1.10 Functions of a Complex Variable

… ► … ►

Analytic Functions

… ►

§1.10(vi) Multivalued Functions

… ►

§1.10(vii) Inverse Functions

… ►

§1.10(xi) Generating Functions

…

28: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

►

§4.37(i) General Definitions

… ►Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

Other Inverse Functions

… ►

§4.37(vi) Interrelations

…

29: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ► … ►Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►

Polynomials

… ►

§16.2(v) Behavior with Respect to Parameters

…

30: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

►

§4.23(i) General Definitions

… ►

Other Inverse Functions

… ►

§4.23(viii) Gudermannian Function

… ►The inverse Gudermannian function is given by …