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Chapter S4: Special Functions and Their Properties
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
An elliptic function is a function that is the inverse of an elliptic integral. An elliptic function is a doubly periodic meromorphic function of a complex variable. All its periods can be written in the form 2mω1 + 2nω2 with integer m and n, where ω1 and ω2 are a pair of (primitive) half‐periods. The ratio τ = ω2/ω1 is a complex quantity that may be considered to have a positive imaginary part, Imτ>0. $ {\text{Im }}\tau> 0. $
The static electromagnetic field
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
are complete elliptic integrals of the first and second kinds, respectively. To compute the magnetic flux density we form B = ∇ × A and find Bϕ = 0, Bz=μI2πF[a2−ρ2−z2(a−ρ)2+z2E(k2)−K(k2)]Bρ=μI2π(zρ)1F[a2−ρ2−z2(a−ρ)2+z2E(k2)−K(k2)].
Introduction to Differential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
where F(∅, k) is the elliptic integral of the first kind with k < 1 (Abramowitz and Stegun, 1964). This integral cannot be evaluated in terms of any known functions, and thus, a numerical table of this integral has been evaluated. The elliptic integral appears naturally in many problems in engineering and science, including the elliptic motions of celestial bodies. This function was studied extensively by A. Legendre. To find the period of pendulum oscillations, we note that when θ = co, the angular velocity is dθldt = 0 and thus ∅ = π/2. Consequently, the period T becomes T=4Lg∫0π/2dϕ1-k2sin2ϕ=4LgF(π/2,k)=4LgK(k) $$ T = 4\sqrt {\frac{L}{g}} \mathop \smallint \limits_{0}^{{\pi /2}} \frac{{d\phi }}{{\sqrt {1 - k^{2} {\text{sin}}^{2} \phi } }} = 4\sqrt {\frac{L}{g}} F(\pi /2,k) = 4\sqrt {\frac{L}{g}} K(k) $$
On the measurement of the bend elastic constant in nematic liquid crystals close to the nematic-to-SmA and the nematic-to-NTB phase transitions
Published in Liquid Crystals, 2021
where and denotes the complete elliptic integral of the first kind