By Brian H. Chirgwin and Charles Plumpton (Auth.)

ISBN-10: 0080131328

ISBN-13: 9780080131320

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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 5

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Prove that the displacement of the other particle at a subsequent time t is 2J . o/1 2 3mn - &mnt sin \ —2 nt 12. A light string, AB, of length 31 is stretched under tension P between two fixed points. Masses 6m and 8m are attached to it at the points X, Y respectively of trisection. The system is at rest until a small transverse velocity V is suddenly given to the particle X. Prove that in the subsequent motion the displacement of 60 A COURSE OF M A T H E M A T I C S the other particle is where n2 — PI ml.

R. h. side of this equation is permissible a n d §1:6 FOURIER SERIES 35 using the integral relations of pp. 1-2, we find π \f{x)}*dx = ^<% + Σ(<£ + %). 13) —π This is ParsevaVs theorem and is true whenever {f(x)}2 is bounded and integrable in the range — π < x < π. Suppose next that the Fourier series for F (x), valid for — π < x < π , is F(x) = %A0 + Σ An cosna; + Σ 1 1 B n sinn re. 14) Then eqns. 14) give by addition F(x) + f(x) =i(A0 + a0) + Σ(Αη 1 + an) cosnx + Σ (Bn + K) *™nx, 1 and by subtraction F(x) - f(x) = i(A0 - a0) + Σ (An - an) cosnx + Σ (Bn ~ 1 1 bn)sinnx.

The fundamental advantage of these transformation methods is that they can lead, usually with little labour, directly to the solutions of the appropriate equations without the introduction of arbitrary constants or arbitrary functions of integration; the initial or boundary conditions are incorporated as the work proceeds. We define the Laplace transform, L {f(t)}, of a function f(t) as the function F(p) given by the equation oo Hf(t)}=F(p)=fe-*f(t)dt. 1) 0 At this stage we take p to be real and positive and assume that the definite integral in eqn.

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A Course of Mathematics for Engineers and Scientists. Volume 5 by Brian H. Chirgwin and Charles Plumpton (Auth.)


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