#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Transforms ---------- " User #4: " Laplace and inverse Laplace transforms " User #5: " => s/[s^2 + (w - 1)^2] (Re s > |Im(w - 1)|) " User #6: " [Gradshteyn and Ryzhik 17.13(33)] " User #7: s :epsilon Real (0, inf) User #8: s Simp(#7) #9: q_ := LAPLACE(COS((w - 1)*t), t, s) User #10: q_ User s --------------- #11: 2 2 Simp(#10) s + (w - 1) #12: "inverselaplace(q_, s, t)" User User #13: " => w/(s^2 - 4 w^2) (Re s > |Re w|) [Gradshteyn and Ryzhik 17.13(84)] " #14: LAPLACE(SINH(w*t)*COSH(w*t), t, s) User w #15: --------------------- Simp(#14) (s + 2*w)*(s - 2*w) User #16: " e^(-6 sqrt(s))/s (Re s > 0) [Gradshteyn and Ryzhik 17.13(102)] " / / 3 \ \ #17: LAPLACE|ERF|---------|, t, s| User \ \ SQRT(t) / / inf / - s*t / 3 \ #18: | #e *ERF|---------| dt Simp(#17) / \ SQRT(t) / 0 User #19: " Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the " User #20: " Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and " User #21: " duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T. " User #22: " Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing " User #23: " Company, 1983, p. 211. First, take the Laplace transform of the ODE " #24: " => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)] " User #25: " where Y(s) is the Laplace transform of y(t) " User #26: Y(t) := User User //d \2 #27: q_ := LAPLACE||--| Y(t) + Y(t) = 4*(STEP(t - 1) - STEP(t - 2)), \\dt/ \ t, s| / #28: q_ User Simp(#28) inf inf -s / - s*t / - s*t 4*#e #29: / #e *Y''(t) dt + / #e *Y(t) dt = -------- - 0 0 s - 2*s 4*#e ----------- s User #30: " Now, solve for Y(s) and then take the inverse Laplace transform " User #31: " => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)] " User #32: " => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)} " #33: y := User User #34: " What is the Laplace transform of an infinite square wave? " User #35: " => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity ) " #36: " [Sanchez, Allen and Kyner, p. 213] " User / inf n \ #37: LAPLACE|1 + 2*SUM (-1) *STEP(t - n*a), t, s| User \ n=1 / Simp(#37) inf / - s*t inf / COS(pi*n) SIGN(a*n - t)*COS(pi*n) \ #38: 2*| #e *SUM |----------- - -------------------------| dt / n=1 \ 2 2 / 0 1 + --- s User #39: " Fourier transform => sqrt(2 pi) delta(z) [Gradshteyn and Ryzhik 17.23(1)] " User inf 1 / #i*z*x #40: FOURIER_TRANSFORM(f, x, z) := ------------*/ f*#e dx SQRT(2*pi) -inf #41: FOURIER_TRANSFORM(1, x, z) User #42: ? Simp(#41) User #43: " => e^(-z^2/36) / [3 sqrt(2)] [Gradshteyn and Ryzhik 17.23(13)] " 2 #44: FOURIER_TRANSFORM(EXP(- 9*x ), x, z) User inf / 2 | - 9*x #45: SQRT(2)*/ #e *COS(x*z) dx Simp(#44) 0 ----------------------------------- SQRT(pi) User #46: " => sqrt(2 / pi) (9 - z^2)/(9 + z^2)^2 [Gradshteyn and Ryzhik 17.23(11)] " #47: FOURIER_TRANSFORM(|x|*EXP(- 3*|x|), x, z) User 2 SQRT(2)*(9 - z ) #48: -------------------- Simp(#47) 2 2 SQRT(pi)*(z + 9) #49: " Mellin transforms " User User #50: " => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)] " inf / s - 1 #51: MELLIN_TRANSFORM(f, x, s) := / f*x dx User 0 #52: s :epsilon (0, 1) User #53: s Simp(#52) / 1 \ #54: MELLIN_TRANSFORM|-------, x, s| User \ 1 - x / inf / s - 1 | x #55: - | -------- dx Simp(#54) / x - 1 0 User #56: " => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1) " #57: " [Gradshteyn and Ryzhik 17.43(16)] " User / BESSEL_J(3, x) \ MELLIN_TRANSFORM|----------------, x, s| #58: | 3 | User \ x / Memory Full / JN(3, x) \ MELLIN_TRANSFORM|----------, x, s| #59: | 3 | User \ x / Memory Full User #60: " Z transforms. See _CRC Standard Mathematical Tables_, Twenty-first Edition, " #61: " The Chemical Rubber Company, 1973, p. 518. " User #62: " Z[H(t - m T)] => z/[z^m (z - 1)] " User #63: STEP(t - 3) User SIGN(t - 3) 1 #64: ------------- + --- Simp(#63) 2 2 #65: STEP(t - m) User 1 SIGN(m - t) #66: --- - ------------- Simp(#65) 2 2 #67: " ---------- Quit ---------- " User