Using Euler's Formula to Solve Laplace Transforms

Complex Integrals and the Laplace Transform There are a few computations for which the complex exponential is very nice to use. We’ll see a few here, but first we show that integrating the complex exponential works out the same as the real exponential: Theorem: As in the real case, Proof: Z e(bi)t dt = Z e(bt)i dt = Z 1 1 eat dt = eat , we have: e(bi)t dt = e(bi)t a bi Z Z cos(bt) + i sin(bt) dt = Z cos(bt) dt + i Z sin(bt) dt = 1 i sin(bt) − i cos(bt) sin(bt) − cos(bt) = b b b And 1 (bt)i cos(bt) + i sin(bt) i − sin(bt) + i cos(bt) sin(bt) − i cos(bt) e = · = = bi bi i −b b Therefore, these quantities are the same. Theorem: As in the real case the previous theorem works for a full complex exponential: Z 1 e(a+bi)t e(a+bi)t dt = (a + bi) You can work this out, but it is more complicated since we’ll need to do integration by parts twice for each integral. It is a nice exercise to try out when you have a little time. Theorem: As in the proof of the last theorem, a corollary is the following: 1 e cos(bt) dt = Re e(a+ib)t e dt = Re a + ib  Z   Z 1 (a+ib)t at (a+bi)t e e sin(bt) dt = Im e dt = Im a + ib Proof: The proof just uses the definition and the earlier theorem. Here we show the proof for the cosine; the proof for the sine is very similar. By Euler’s Formula: Z Z at (a+bi)t    eat cos(bt) + ieat sin(bt) = e(a+ib)t Therefore, we can integrate both sides: Z at e cos(bt) dt + i Z at e sin(bt) dt = Z e(a+ib)t dt = 1 e(a+ib)t a + ib Now, take the real part of each expression: Z eat cos(bt) dt = Re Z  e(a+ib)t dt = Re  1 e(a+ib)t a + ib  Of course, after this we need to actually compute this last expression. Recall that division by complex numbers is defined as: 1 1 a − ib a b = · = 2 −i 2 2 a + ib a + ib a − ib a +b a + b2 1 Worked Example: 1. Use complex exponentials to compute Z e2t cos(3t) dt. SOLUTION: We note that e2t cos(3t) = Re(e(2+3i)t ), so: 1 e(2+3i)t 2 + 3i Simplifying the term inside the parentheses and multiplying out the complex terms: Z e2t cos(3t) dt = Re   2 − 3i e (cos(3t) + i sin(3t)) = 4+9     2 3 3 2 2t e cos(3t) + sin(3t) + i − cos(3t) + sin(3t) 13 13 13 13 2t   Therefore, Z e2t cos(3t) dt = e2t  2 3 cos(3t) + sin(3t) 13 13  In fact, we get the other integral for free: Z e2t sin(3t) dt = e2t  −3 2 cos(3t) + sin(3t) 13 13  2. Use complex exponentials to compute the Laplace transform of cos(at): SOLUTION: Note that cos(at) = Re(e(at)i ) L(cos(at)) = Z ∞ e −st Z ∞ −st (ai)t e cos(at) dt = Re e  dt = 0 0 t→∞ −1 e−(s−ai)t (s − ai) 0 t=0 What happens to our expression as t → ∞? The easiest way to take the limit is to check the magnitude (see if it is going to zero): Z ∞ Re  e−(s−ai)t dt = Re −1 −st (ai)t −1 e e · e−st · e(ai)t = s − ai s − ai Now, the first term is a constant with respect to t, and e(at)i is a point on the unit circle (see the exercises). Therefore, the magnitude of the integral depends solely on e−st , where s is any real number so that e−st is the standard real exponential function. For s > 0, the function e−st → 0 as t → ∞, so that the entire expression will also go to zero if s > 0: −1 e−(s−ai)t = 0 lim t→∞ (s − ai) and the Laplace transform is:  L(cos(at)) = Re 0 − −1 s + ai = Re 2 s − ai s + a2    = s s 2 + a2 As a side remark, we get the Laplace transform of sin(at) for free since it is the imaginary part. 2 Homework Addition to Section 6.1 1. Show that eiat is on the unit circle- That is, show that its magnitude is 1 (for any value of a, t). 2. Use complex exponentials to compute Z e−2t sin(3t) dt. 3. Use complex exponentials to compute the Laplace transform of sin(at). 4. Use complex exponentials to compute the Laplace transform of eat sin(bt) and eat cos(bt) (compare to exercises 13, 14). 5. Prove that et goes to infinity faster than any polynomial. You can do that by showing tn =0 t→∞ et lim 6. Use the “Racetrack Principle”1 to prove that ln(t) < t for all t > 1. 7. Show that, if f (t) is bounded (that is, there is a constant A so that |f (t)| ≤ A for all t), then f is of exponential order (do this by finding K, a and M from the definition). 8. If the function is of exponential order, find the K, a and M from the definition. Otherwise, state that it is not of exponential order. Something that may be handy from algebra: A = eln(A) . 2 (a) sin(t) (d) et (b) tan(t) (e) 5t (c) t3 (f) tt 9. Use complex exponentials to find the Laplace transform of t sin(at). The Racetrack Principle says that if f (a) ≤ g(a) and f 0 (x) < g 0 (x) for all x in some interval (a, b), then f (x) < g(x) for all x in (a, b) (think of f (x) and g(x) as the positions of two cars on a track, so that the derivatives are their velocities). 1 3