DLMF: Notations F

Notations F

♦*♦A♦B♦C♦D♦E♦F♦G♦H♦I♦J♦K♦L♦M♦N♦O♦P♦Q♦R♦S♦T♦U♦V♦W♦X♦Y♦Z♦

$F_{n}$: Fibonacci number; §26.11
$\mathop{F_{D}\/}\nolimits$: Lauricella’s multivariate hypergeometric function; §19.15
$F(x)$: Fourier transform; (1.14.1)
$\mathop{F\/}\nolimits\!\left(z\right)$: Dawson’s integral; (7.2.5)
$\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$: Fresnel integral; (7.2.6)
$\mathsf{F}(z-1)=\mathop{\psi\/}\nolimits\!\left(z\right)$: notation used by Pairman (1919); §5.1
(with $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function)
$\mathop{\mathit{f}\/}\nolimits\!\left(x\right)$: Euler’s reciprocal function; (27.14.2)
$\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$: auxiliary function for sine and cosine integrals; (6.2.17)
$\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$: auxiliary function for Fresnel integrals; (7.2.10)
$F_{{\nu}}(z)=\mathop{\mathrm{Me}_{{\nu}}\/}\nolimits\!\left(z,q\right)$: notation used by Abramowitz and Stegun (1964, Chapter 20); ¶ ‣ §28.1
(with $\mathop{\mathrm{Me}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function)
$F_{s}(x)$: Fermi–Dirac integral; (25.12.14)
$F_{c}(x)$: Fourier cosine transform; (1.14.9)
$F_{s}(x)$: Fourier sine transform; (1.14.10)
$F_{p}(z)$: terminant function; (8.22.1)
$f_{{\mathit{e},m}}(h)$: joining factor for radial Mathieu functions; §28.22(i)
$f_{{\mathit{o},m}}(h)$: joining factor for radial Mathieu functions; §28.22(i)
$F(\phi\backslash\alpha)=\mathop{F\/}\nolimits\!\left(\phi,k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind)
$\mathop{F\/}\nolimits\!\left(\phi,k\right)$: Legendre’s incomplete elliptic integral of the first kind; (19.2.4)
$\mathop{F\/}\nolimits\!\left(x,s\right)$: periodic zeta function; (25.13.1)
$\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$: regular Coulomb radial function; (33.2.3)
$\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right)$: hypergeometric function; §15.1
$\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$: hypergeometric function; (15.2.1)
$\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop c};z\right)$: Olver’s hypergeometric function; §15.1
$\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$: Olver’s hypergeometric function; (15.2.2)
$f(\epsilon,\ell;r)=\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$: notation used by Greene et al. (1979); ¶ ‣ §33.1
(with $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function)
$\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$: regular Coulomb function; (33.14.4)
$f^{{(0)}}(\epsilon,\ell;r)=\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$: notation used by Greene et al. (1979); ¶ ‣ §33.1
(with $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function)
$\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a\atop b};\mathbf{T}\right)$: confluent hypergeometric function of matrix argument (first kind); §35.6(i)
$\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left(a;b;\mathbf{T}\right)$: confluent hypergeometric function of matrix argument (first kind); §35.6(i)

$\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left(a;b;\mathbf{T}\right)$: confluent hypergeometric function of matrix argument (first kind); §35.1
$\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{T}\right)$: hypergeometric function of matrix argument; §35.7(i)
$\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b;c;\mathbf{T}\right)$: hypergeometric function of matrix argument; §35.1
$\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b;c;\mathbf{T}\right)$: hypergeometric function of matrix argument; §35.7(i)
$\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b;c;z\right)$: hypergeometric function; §15.1
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$: generalized hypergeometric function; (16.2.1)
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$: generalized hypergeometric function; §16.5
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$: generalized hypergeometric function; §16.5
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$: generalized hypergeometric function; (16.2.1)
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{p}\atop b_{1},b_{2},\dots,b_{q}};\mathbf{T}\right)$: generalized hypergeometric function of matrix argument; §35.8(i)
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$: generalized hypergeometric function; §16.5
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$: generalized hypergeometric function; (16.2.1)
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)$: generalized hypergeometric function; §16.5
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)$: generalized hypergeometric function; (16.2.1)
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(a_{1},a_{2},\dots,a_{p};b_{1},b_{2},\dots,b_{q};\mathbf{T}\right)$: generalized hypergeometric function of matrix argument; §35.8(i)
$\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(a_{1},a_{2},\dots,a_{p};b_{1},b_{2},\dots,b_{q};\mathbf{T}\right)$: generalized hypergeometric function of matrix argument; §35.1
$\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,b;c;z\right)$: Olver’s hypergeometric function; §15.1
$\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$: scaled (or Olver’s) generalized hypergeometric function; (16.2.5)
$F(a,b;t:q)$: alternative notation for specialization of $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits$ ; Fine (1988); §17.1
$\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$: Appell function; (16.13.1)
$\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)$: Appell function; (16.13.2)
$\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)$: Appell function; (16.13.3)
$\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$: Appell function; (16.13.4)
$\mathop{\mathrm{Fc}_{{m}}\/}\nolimits\!\left(z,h\right)$: Mathieu function; §28.26(i)
$\mathop{\mathrm{Fe}_{{n}}\/}\nolimits\!\left(z,q\right)$: modified Mathieu function; (28.20.6)
$\mathop{\mathrm{fe}_{{n}}\/}\nolimits\!\left(z,q\right)$: second solution, Mathieu’s equation; (28.5.1)
$\mathrm{Fey}_{n}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{{\mathit{e},n}}(h)\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)\mathop{{\mathrm{Mc}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$: notation used by Arscott (1964b), McLachlan (1947); ¶ ‣ §28.1
(with $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function and $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function)
$\mathop{\mathrm{Fs}_{{m}}\/}\nolimits\!\left(z,h\right)$: Mathieu function; §28.26(i)