Notations F

$F_{D}$: Lauricella’s multivariate hypergeometric function; §19.15
$F_{\NVar{n}}$: Fibonacci number; §26.11
$\mathit{f}\left(\NVar{x}\right)$: Euler’s reciprocal function; (27.14.2)
$\mathrm{f}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals; (7.2.10)
$\mathrm{f}\left(\NVar{z}\right)$: auxiliary function for sine and cosine integrals; (6.2.17)
$F\left(\NVar{z}\right)$: Dawson’s integral; (7.2.5)
$\mathcal{F}\left(\NVar{z}\right)$: Fresnel integral; (7.2.6)
$\mathsf{F}(\NVar{z-1})=\psi\left(z\right)$: notation used by Pairman (1919); §5.1
(with $\psi\left(\NVar{z}\right)$ : psi (or digamma) function)
$f_{\mathit{e},\NVar{m}}(\NVar{h})$: joining factor for radial Mathieu functions; §28.22(i)
$F_{\NVar{\nu}}(\NVar{z})=\operatorname{Me}_{\nu}\left(z,q\right)$: notation used by Abramowitz and Stegun (1964, Chapter 20); §28.1
(with $\operatorname{Me}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function)
$f_{\mathit{o},\NVar{m}}(\NVar{h})$: joining factor for radial Mathieu functions; §28.22(i)
$F_{\NVar{p}}\left(\NVar{z}\right)$: terminant function; (2.11.11)
$F_{\NVar{s}}(\NVar{x})$: Fermi–Dirac integral; (25.12.14)
$\mathscr{F}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Fourier transform; (1.14.1)
$\mathscr{F}_{\mkern-3.0muc}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Fourier cosine transform; (1.14.9)
$\mathscr{F}_{\mkern-2.0mus}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Fourier sine transform; (1.14.10)
$\mathscr{F}\left(\NVar{u}\right)$: Fourier transform of a tempered distribution; (1.16.35)
$F(\NVar{\phi}\backslash\NVar{\alpha})=F\left(\phi,k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind)
$F\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the first kind; (19.2.4)
$F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$: regular Coulomb radial function; (33.2.3)
$F\left(\NVar{x},\NVar{s}\right)$: periodic zeta function; (25.13.1)

$F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function; (15.2.1)
$\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function; (15.2.2)
$f(\NVar{\epsilon},\NVar{\ell};\NVar{r})=s\left(\epsilon,\ell;r\right)$: notation used by Greene et al. (1979); item Greene et al. (1979):
(with $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function)
$f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$: regular Coulomb function; (33.14.4)
${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$: $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function; §16.2
${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ or ${{}_{1}F_{1}}\left({\NVar{a}\atop\NVar{b}};\NVar{\mathbf{T}}\right)$: confluent hypergeometric function of matrix argument (first kind); §35.6(i)
${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function; §16.2
${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{\mathbf{T}}\right)$ or ${{}_{2}F_{1}}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{\mathbf{T}}\right)$: Gaussian hypergeometric function of matrix argument; (35.7.1)
${{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: Olver’s hypergeometric function; (15.2.2)
${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$: alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function; §16.2
${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$: scaled (or Olver’s) generalized hypergeometric function; (16.2.5)
${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$: generalized hypergeometric function of matrix argument; (35.8.1)
$f^{(0)}(\NVar{\epsilon},\NVar{\ell};\NVar{r})=f\left(\epsilon,\ell;r\right)$: notation used by Greene et al. (1979); item Greene et al. (1979):
(with $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function)
$F(\NVar{a},\NVar{b};\NVar{t}:\NVar{q})$: alternative notation for specialization of ${{}_{2}\phi_{1}}$ ; Fine (1988); §17.1
${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$: first Appell function; (16.13.1)
${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$: second Appell function; (16.13.2)
${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$: third Appell function; (16.13.3)
${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$: fourth Appell function; (16.13.4)
$\operatorname{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function; (28.20.6)
$\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: second solution, Mathieu’s equation; (28.5.1)
$\mathrm{Fey}_{\NVar{n}}(\NVar{z},\NVar{q})=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{e}% ,n}(h)\operatorname{ce}_{n}\left(0,q\right){\operatorname{Mc}^{(2)}_{n}}\left(% z,h\right)$: notation used by Arscott (1964b), McLachlan (1947); §28.1
(with $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter and ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function)

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Notations F

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