[CHAP. For simple poles the calculation of the residue is of particular simplicity since it reduces to a-, = lim (z - a) f (z) (21) z- a RESIDUE THEOREM If f (z) is analytic in a region `R except for a pole of order n at z =a and if C is any simple closed curve in `R containing z = a, then f (z) has the form (19). 7-6 the poles. Theorem 1-1. 17 + 19i in all cases 80. Ans. 8-4 TWO AND THREE DIMENSIONAL PROBLEMS Many of the above partial differential equations can be generalized to apply to problems in two and three dimensions. Since { e-3t) = s +1 3 we can write e-st UNIQUENESS OF INVERSE LAPLACE TRANSFORMS. 10 = or (z - 1)3 c Hence 5z2-3z+2dz (z -1)'; = fC = 101ri dz (z - 1)3 +7 (z - 1)3 Then 5(z - 1)2 + 7(z - 1) + 4 5 fzdz-1 5z2-3z+2 dz = c 5z2 - 3z + 2 = 5(z - 1)2 + 7(z -1) + 4. The brachistochrone problem is that of finding the shape of a frictionless wire in a vertical plane, as shown in Fig. This represents one branch of the double-valued function w = V-z-. More generally, we refer to the series I ak (z - a)k as a Laurent series k=- where the terms with k < 0 constitute the principal part. Then to = w = 1 [corresponding to A and P in Figures 5-18(a) and (b)]. 8] 225 The general solution of the differential equation is = cl etc/a + c2 a - va/a From the condition on boundedness, we must have cl = 0. 7-4 We have dF' - F" (o) - q, cos sin 0 02 (1) THE COMPLEX INVERSION FORMULA CHAP. 68.
According to Newton's law which states that the net force acting on m is equal to the mass times the acceleration, the equation of motion is 2X m tt = -kX (b) Fig. (b) aie/3 Prove Cauchy's integral formulas. If (z) I < M, and L is the length of the path C. CAUCHY'S THEOREM Let C be a simple closed curve. Laguerre polynomials Ln(t) = n dtn (tne-t), 255 n = 0, 1, 2, .. . 109. 8 x > 0, t > 0 U(0, t) = UO, I U(x, t) 1 < M where the last condition expresses the requirement that the temperature is bounded for all x and t. Taking Laplace transforms, we find au - U(x, 0) = where or kd 2"2 = ,J {U(0, t)} U(018) d2x2 = -ku=0 (1) Uo s (2) and u = u(x,s) is required to be bounded. 27r 27r = lim.f (a + ee{e) do E- 0 0 But since f (z) is analytic, i fo f (a) do = 27ri f (a) and the required result follows.
(b) Illustrate by using f (Z) = z2. Then F'(t) = a cos at, we have a J {F"(t)} 92.C {F(t)} - s F(0) X {- a2 sin at) = or Hence - F'(0) 92 ^e {sin at} - s (0) - a = -a2.C {sin at) i.e. Thus In z is a many-valued function. Calculus for Busineed. Work Problem 55 if 7(t) = F° sin wt, treating the two cases: (a) w # k/m, the physical significance of each case. By trial we find z = 2 is a root. hundreds of thousands of students improve their test scores and final grades with these indispensable study guides. These two possible approaches show that the limit depends on the manner in which Az - 0, that the derivative does not exist; i.e. Show that the deflection at any point is Y(x) _ WO x2(1 - x)2 24E1 76. 8-5 APPLICATIONS TO BOUNDARY-VALUE PROBLEMS 222 at = k U(x, 0) = 0, a2X2 [CHAP. Functions of exponential order. Solve the partial differential equation a2 2 at2 - 4 ax2 + Y 16x + 20 sin x subject to the conditions Y(0, t) = 0, Yt (x, 0) = 0, Y(7r, t) = 167r, = 16x + 12 sin 2x - 8 sin 3x Y(x, 0) Taking Laplace transforms, we find s2y 18x = - 8 Y(x, 0) - Yt (x, 0) - 4 dx2 + y + 20 sin x (1) s or, on using the given conditions, d2y dx2 - 1 4 -4(82 + 1)x = (s2 + 1)y 8 Y(0' s) = 0, - 5 sin x - 3s sin 2x + 2a sin 3x 8 (2) 167, y(7F, 8) = (3) s A particular solution of (2) has the form ax + b sin x + c sin 2x + d sin 3x = yp (4) Then substituting and equating coefficients of like terms, we find the particular solution yp s + _ 12s sin 2x 20 sin x 16x = S(82+5) + 82+17 8s sin 3x S2+37 (5) The general solution of the equation (2) with right hand side replaced by zero [i.e. Frank Ayres Theory & Problems of Differential Equations Schaum Publishing Co. 1952 Acrobat 7 Pdf 10.5 Mb. 11 3 SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS In the following list of theorems we assume, unless otherwise stated, that all functions satisfy the conditions of Theorem 1-1 so that their Laplace transforms exist. Multiplication by tn. (b) Show that the Cauchy-Riemann equations hold at all points in the finite z plane. 0 148. This book is designed for use as a supplement to all current standard texts or as a textbook for a formal course in Laplace transform theory and applications. A further modification takes place when some prescribed time-varying external force f(t) also acts on m. In such case the equation of motion is z M dX = -kX - 8 dt + f(t) or mX" + /3X' + kx = f(t) (7) By using Laplace transforms to solve equations (5), (6) or (7)- subject to various appropriate initial conditions of physical interest, the displacement X(t) can be found.
... integral equations, difference equations and boundary-value problems. z = 1 is a pole of order 2, or double pole. Modulus r = (-1)2 + (0)2 = 1.
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