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

Legendre elliptic integrals