By Reinhard Diestel
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Additional resources for A Fourier Analysis And Its Applications
Heaviside’s function is connected with the δ function via the formula t H(t) = δ(u) du. −∞ A very bold diﬀerentiation of this formula would give the result H (t) = δ(t). 5) 30 2. 5) is reasonable for t = 0. What is new is that the “derivative” of the jump discontinuity of H should be considered to be the “pulse” of δ. In fact, this assertion can be given a completely coherent background; this will be done in Chapter 8. , it has a continuous derivative, and if in addition ϕ is zero outside some ﬁnite interval, the following calculation is clear: ∞ ∞ ϕ (t)H(t) dt = −∞ ϕ (t) dt = ϕ(t) 0 ∞ t=0 = 0 − ϕ(0) = −ϕ(0).
We recognize the transform as a derivative: g(s) = − 1 d . 2 and the known transform of the sine we get g(t) = t sin t. 11. Solve the initial value problem y + 4y + 13y = 13, y(0) = y (0) = 0. Solution. Transformation gives (s2 + 4s + 13)y = 13 13 ⇐⇒ y = . s s (s + 2)2 + 9 50 3. Laplace and Z transforms Expand into partial fractions: y= s+4 1 s+2 1 − = − − s (s + 2)2 + 9 s (s + 2)2 + 9 2 3 · 3 . (s + 2)2 + 9 The solution is found to be y(t) = 1 − e−2t (cos 3t + 2 3 sin 3t) H(t). (Here we have multiplied the result by a Heaviside factor, to indicate that we are considering the solution only for t ≥ 0.
If these things are quite new, the reader is also advised to ﬁnd more exercises in textbooks where complex numbers are treated. 1 Compute the numbers eiπ/2 , e−iπ/4 , e5πi/6 , eln 2−iπ/6 . , that f (z+2πi) = f (z) for all z. 3 Find a formula for cos 3t, expressed in cos t, by manipulating the identity 3 e3it = eit . 4 Prove the formula sin3 t = z 3 4 sin t − 1 4 sin 3t. 5 Show that if |e | = 1, then z is purely imaginary. 6 Prove the de Moivre formula: (cos t + i sin t)n = cos nt + i sin nt, n integer.
A Fourier Analysis And Its Applications by Reinhard Diestel