amplitude%20%28am%29%20function

SearchAdvancedHelp

(0.004 seconds)

21—30 of 965 matching pages

21: 25.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,m,n$	nonnegative integers.
…
primes	on function symbols: derivatives with respect to argument.

►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

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 $D_{\nu }\left(z\right)$. …

23: 17.1 Special Notation

§17.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►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: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

►

§4.37(i) General Definitions

… ►Each of the six functions is a multivalued function of $z$. … ►

Other Inverse Functions

… ►

§4.37(vi) Interrelations

…

25: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►

§11.10(v) Interrelations

… ►

§11.10(vi) Relations to Other Functions

… ► … ►

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

…

26: 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 …

27: 23.2 Definitions and Periodic Properties

… ►

§23.2(i) Lattices

… ► … ►

§23.2(ii) Weierstrass Elliptic Functions

… ► ►

§23.2(iii) Periodicity

…

28: 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

…

29: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►

§12.14(vii) Relations to Other Functions

►

Bessel Functions

… ►

Confluent Hypergeometric Functions

… ►

§12.14(x) Modulus and Phase Functions

…

30: 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 $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

…