Fungrim entry: da16db

R_F\!\left(0, y, z\right) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)}} \, d\theta

Assumptions:

y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

TeX:

R_F\!\left(0, y, z\right) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)}} \, d\theta

y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("da16db"),
    Formula(Equal(CarlsonRF(0, y, z), Integral(Div(1, Sqrt(Add(Mul(y, Pow(Cos(theta), 2)), Mul(z, Pow(Sin(theta), 2))))), For(theta, 0, Div(Pi, 2))))),
    Variables(y, z),
    Assumptions(And(Element(y, CC), Element(z, CC), Greater(Re(y), 0), Greater(Re(z), 0))))

Topics using this entry

Carlson symmetric elliptic integrals