# Fungrim entry: b7cfb3

$F\!\left(\phi, 1\right) = \begin{cases} \log\!\left(\frac{1 + \sin(\phi)}{\cos(\phi)}\right), & \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2} \;\mathbin{\operatorname{and}}\; \phi \notin \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\\operatorname{sgn}(\phi) \infty, & \phi \in \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}$
Assumptions:$\phi \in \mathbb{C}$
TeX:
F\!\left(\phi, 1\right) = \begin{cases} \log\!\left(\frac{1 + \sin(\phi)}{\cos(\phi)}\right), & \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2} \;\mathbin{\operatorname{and}}\; \phi \notin \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\\operatorname{sgn}(\phi) \infty, & \phi \in \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}

\phi \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
IncompleteEllipticF$F\!\left(\phi, m\right)$ Legendre incomplete elliptic integral of the first kind
Log$\log(z)$ Natural logarithm
Sin$\sin(z)$ Sine
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Re$\operatorname{Re}(z)$ Real part
Sign$\operatorname{sgn}(z)$ Sign function
Infinity$\infty$ Positive infinity
UnsignedInfinity${\tilde \infty}$ Unsigned infinity
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("b7cfb3"),
Formula(Equal(IncompleteEllipticF(phi, 1), Cases(Tuple(Log(Div(Add(1, Sin(phi)), Cos(phi))), And(LessEqual(Div(Neg(Pi), 2), Re(phi), Div(Pi, 2)), NotElement(phi, Set(Div(Neg(Pi), 2), Div(Pi, 2))))), Tuple(Mul(Sign(phi), Infinity), Element(phi, Set(Div(Neg(Pi), 2), Div(Pi, 2)))), Tuple(UnsignedInfinity, Otherwise)))),
Variables(phi),
Assumptions(Element(phi, CC)))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC