By Gray A., Mathews G.B.

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**Extra resources for A treatise on Bessel functions and their applications to physics**

**Sample text**

1. Find s, and ﬁnd the parametric description of the curve given by the arc length. It is proved in Diﬀerential Geometry that any plane curve has a curvature κ(t) = {ez × r (t)} · r (t) , r (t) 3 where we let the plane of the curve be the (X, Y )-plane in the space. 2. Prove that κ is proportional to s for K. The curve under consideration has many names: the clothoid, Euler’s spiral, Cornu’s spiral. Remark. e. Ancient Greek: κλωθω = I spin. ♦ A Parametric description with respect to arc length, curvature.

1) Alternatively, 2 + 2 d dϕ = √ ϕ 2e , hence 1 K 4) Here ds = 0 √ √ ϕ 2 e dϕ = 2(e − 1). ϕ ϕ d = a · sin2 · cos , so dϕ 3 3 2 + d dϕ 2 =a sin6 ϕ ϕ ϕ ϕ + sin4 · cos2 = a · sin2 , 3 3 3 3 thus 3π K ds = 0 a sin2 ϕ dϕ = 3a 3 π 0 sin2 t dt = 3a 2 π 0 (1 − cos 2t) dt = 3aπ . 1. 3 Below are given some space curves by their parametric descriptions x = r(t), t ∈ I. Express for each of the curves there parametric description with respect to arc length from the point of the parametric value t0 . 1) The curve r(t) = (cos t, sin t, ln cos t), from t0 = 0 in the interval I = 0, π .

A Parametric description by the arc length. D Find s (t) = r (t) and then s = s(t) and t = τ (s), where we integrate from t0 . Finally, insert in x = r(t) = r(τ (s)). com 39 Calculus 2c-7 Arc lengths and parametric descriptions by the arc length hence t s(t) = 0 1 ln 2 1 + sin u 1 − sin u t 0 cos u du = 1 − sin2 u t = 0 1 ln 2 t 0 1 + sin t 1 − sin t 1 1 + 1 + sin u 1 − sin u 1 2 cos u du . 2. Then 1 + sin t = e2s , 1 − sin t dvs. Notice that it follows from t ∈ 0, cos t = e2s − 1 = tanh s, e2s + 1 sin t = s ≥ 0.

### A treatise on Bessel functions and their applications to physics by Gray A., Mathews G.B.

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