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Date Created: 09/28/15
Math 241 Spring 2005 Review Sheet for Second Midterm Be able to o Solve the wave equation not the heat equation using Laplace transforms Know how to invert the standard Laplace transforms including those of the form e 5Fsi 0 Compute the Fourier transform Fourier sine transform and Fourier cosine transform of a simple function 0 Use the three Fourier transforms to solve the wave heat or Laplace equations 7 If 700 lt z lt 00 use the complex Fourier transform 7 If 0 S I lt 00 with u0t speci ed use the Fourier sine transform 7 If 0 S I lt 00 with 0t speci ed use the Fourier cosine transform 0 Perform basic algebra with complex numbers including multiplication division conjugation absolute values arguments powers and roots 0 Identify basic sets of complex numbers and describe how lines get transformed by complex functions Prove that a limit or derivative of a complex function does not exist by showing that limits horizontally and vertically are different 0 Verify that a function is differentiable possibly only at certain points using the CauchyRiemann equations 0 Compute the conjugate harmonic function vz y given a harmonic function uz 1 1 0 Compute 6 Ln 2 ln 2 sin 2 cos 2 tan 2 sin 2 cos 2 and tan 1 2 for any complex number 2 0 Compute a simple contour integral by hand using a speci c parametrization ie without using Cauchy Goursat or the Cauchy Integral Formula 0 Use the principle of path independence to compute nonclosed contour integrals of analytic functions by nding complex antiderivatives 0 Use the Cauchy Integral Formulas to compute closed contour integrals of analytic functions 0 Determine whether a complex series converges absolutely converges or diverges by applying the nthterm test the p series test for ip and the geometric series test for 0 Determine the radius of convergence of a complex power series with given coef cients For simple examples determine which points on the circle of convergence the power series converges 0 Determine what the radius of convergence must be for a given function7s Taylor series at any point by computing the distance to the nearest singularity 0 Use standard power series such as for ll 7 2 6 sin 2 and cos 2 to derive the Taylor series of functions like 2 l 7 z2 62 cos 2 etc around any given complex number 20 This may involve algebra or termby term calculus Important concepts you should understand 0 The differences between Laplace and Fourier transforms 7 Laplace transforms are transforms in the time variable and are used only for problems with 0 S t lt 00 7 Fourier transforms are transforms in the space variable and are used only for problems with 0 S I lt 00 or 700 lt z lt 00 Thus for example to solve the Laplace equation on an in nite strip you transform with respect to the in nitedomain variable only 7 Laplace transforms can be used no matter what the domain of z is 7 Fourier transforms have simple inversion formulas unlike Laplace transforms 7 Many equations can be solved using either technique 7 Laplace transforms give simple answers to the wave equation dif cult answers for the heat equation 7 Fourier transforms give answers that are of medium complexity to any equation 7 Neither transform technique changes at all if the equation is nonhomogeneous 7 Fourier transforms will only work if the solution is assumed to go to zero as the space variable approaches in nity This condition is often assumed implicitly for the standard PDEs o The difference between argz and Arg 2 as well as between lnz and Ln 2 0 Why rules such as Ln 2122 Ln 21 Ln 22 may fail but ln 2122 ln 21 ln 22 are still true Where the CauchyRiemann equations come from and why they imply that the real and imaginary parts of any complex function are harmonic o The de nition of the complex derivative and why many complex functions do not have derivatives The difference between being di erentiable at a point and being analytic at the point For example if 52 then f is not analytic at 2 0 but fO 0 o q39 pl vs 39p of a domain A simplyconnected domain is one in which any closed curve can be shrunk down to a point while still being inside the domain A multiplyconnected domain is one for which there is some hole that prevents this shrinking o How to compute a contour integral when the contour contains multiple singularities or when the contour changes orientation You should be able to do all of the Core Problems for Sections 15271547 17171787 1817184 and 1917192 In addition you should be able to do the following review exercises Many of these exercises combine several topics from di erent sections these are excellent practice for the exam 0 Chapter 15 pp 7657766 6 using Laplace transforms l 4 7 8 9 ll 13 using Fourier transforms 0 Chapter 17 pp 8247825 172022724263073137740 Note the review problems here inexplicably do not include much of the later portions of Chapter 17 You may want to practice some of the following problems 7 175 1728 7 176 23738 7 177 1714 and 21730 7 178 1710 0 Chapter 18 pp 8497850 1 377 9714 16725 27728 0 Chapter 19 pp 8867887 172 11713 Math 241 Spring 2005 Review Sheet for First Midterm Be able to o 1211 Determine if functions and 91 are orthogonal on an interval ab possibly with respect to a weight function w z i o 1212 1213 Compute the coefficients of the four possible Fourier series for a function 7 A full Fourier series both sines and cosines for any function de ned on 71010 7 A cosine series for an even function on 71010 or for any function on 0Li 7 A sine series for an odd function on 71010 or for any function on 0 L 7 A full Fourier series both sines and cosines for any function on 0Li 1212 1213 Graph the function to which the Fourier series converges including the case where there is a jump discontinuity in the original function corresponding to the four cases above 7 The periodic extension with period 2p of any function de ned on 71010 7 The even periodic extension with period 2L of any function de ned on 0Li 7 The odd periodic extension with period 2L of any function de ned on 0Li 7 The periodic extension with period L of any function de ned on 0Li o 1214 Compute the coefficients on of a complex Fourier series of a function de ned on 7ppi Also compute their magnitudes 15 ii 0 1215 Put a secondorder differential equation in selfadjoint formi 1215 Find the eigenvalues and eigenfunctions for a Regular Sturm Liouville problem when the differential equation is either a constantcoef cient DE or a CauchyEuler DE If the eigenvalues cannot be found explicitly nd a relation for them as in Example 2 of Section 1214 o 1311 Classify any secondorder linear partial differential equations in two variables as elliptic hyperbolic or para o ic o 1311 Use separation of variables to nd certain solutions of linear partial differential equations of the form Way XIYy o 132 Translate word problems for the heat equation wave equation and Laplace equation into boundary conditions for the appropriate PDEi In particular understand xing temperature vsi insulating a boundary 1313 1314 Solve the heat equation and wave equation for a rodstring with homogeneous boundary conditions and given initial condition s i o 1315 Solve the Laplace equation on a rectangle with prescribed boundary values the Dirichlet problem Be able to identify which variable gives a Sturm Liouville problem with homogeneous boundary values 1316 Find a steadystate solution of a possibly nonhomogeneous heat equation or wave equation with possibly nonhomogeneous boundary conditions Use this steadystate solution to get a homogeneous equation for vzt 111775 1317 Find an expansion for a function in terms of orthogonal functions that arise in SturmLiouville problems from separation of variables 1318 Solve a twodimensional heat equation with zero temperature on the boundary or a twodimensional wave equation with zero displacement on the boundary using double sine series 0 1411 Solve the Laplace equation on a disc or a portion of a disc or an annulus with given boundary conditions Important concepts you should understand 0 Orthogonality of functions is used to obtain simple formulas for the coefficients in Fourier and other series Review the relationship between formulas 7 and 8 in Section 121 and the computations in Section 137 o A Fourier series converges at every point of continuity In case of a jump discontinuity the Fourier series will converge to the average value A jump discontinuity may occur in the periodic extension even if it does not occur in the original interval 71010 0 There are four different types of Fourier series depending on whether one is working on a symmetric interval or not and whether one knows anything about the function evenness or oddness or not The formulas for the coefficients are all similar but not the same Make sure you donlt get them confused The properties of a Sturm Liouville problem 7 there are in nitely many eigenvalues A1 A2 etc increasing to in nity 7 the eigenfunctions are all orthogonal on the interval with respect to the weight function These facts are used in Section 137 to study more complicated boundary conditions for which the eigenvalues cannot be found explicitly eg the solutions of tan 7A We can still get a complete formula for the solution even without knowing A o The heat equation u Kum is a typical parabolic equation 7 To get a unique solution we must specify an initial condition uz 0 and two boundary conditions 7 The solution will always decay exponentially as t 7gt 00 to some steadystate solution satisfying the boundary conditions 0 The wave equation um 021 is a typical hyperbolic equation 7 To get a unique solution we must specify two initial conditions uz 0 and u z 0 91 and two boundary conditions 7 The solution will never decay it will oscillate for all time like a spring The general solution is a sum of normal modes The Laplace equation um uyy 0 is a typical elliptic equation 7 To get a unique solution we only need to specify boundary conditions either on the edges of a square or on the outer circle of a disc Some old formulas you may need 0 The doubleangle formulas for trig functions For computing orthogonal series expansions o The integration by parts formula For computing Fourier coefficients 0 The general solution of secondorder differential equations with constant coefficients ay by cy 0 Section 33 and the CauchyEuler equation aIQy bzy cy 0 Section 36 For solving Sturm Liouville problems and separation of variables problems You should be able to do all of the Core Problems for Sections 12171257 13171387 and 141 In addition you should be able to do the following review exercises 0 Chapter 12 p 685 17811718 0 Chapter 13 pp 7177718 173 579 11713 0 Chapter 14 p 733 175
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