= Finding the residue with Laurent series expansion for two of the following functions: (1) (2²-2z+2)/(z-2) in the dómain |2 - 1| < 1. (2) (1) in the domain 0 < 2 - 1|< 1. (3) (z-1) sin around z = -2. +2
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- 5. (a) Find the Fourier series for the periodic function defined by f(x) = e*, –n < x < T. Hint: Recall that sinh x = (e* – e-*)/2. (b) Sketch the graph of the sum of this series on the interval -5n s x s 5n. (c) Use the series in (a) to establish the sums 1 1 n' +1 2 tanh t and (-1)" n² +1 1 %3D 2 sinh T d: To what value does the series converge at the points of discontinuity?Determine the Fourier series for the function defined by: -1, -n < x < -· 2 π f(x) = { 1, < x <• -1,Conduct Fourier series expansion for the figure. Along with the solution and how to choose which form to use. (f(x)=a0/2+Σ(an*cosnx+bn*sinnx) or using e^(-jωt)7. Let (a) (b) p(x) = -1-x_ _for-1 < x < 0 +1-x for 0 < x < 1. Find the full Fourier series of p(x) in the interval (-1, 1). Find the first three nonzero terms explicitly. (c) (d) Does it converge pointwise? (e) Does it converge uniformly to p(x) in the interval (-1, 1)? Does it converge in the mean square sense? OR(-1)"(2-5-8. (3n+2))(x-1)2n+2 3n. (n2 +1) The derivative of the power series f(x) = 2 is (- 1)"(2 -5-8 (3n+2))(2n+2)(x- 1)2n+1 3n. (n² +1) %3D n=0 True False Click Save and Submit to save and submit. Click Save All Answers to save all answers.11. (a) Find the Fourier series of the 27-periodic function given on the interval -<< π by f(x) = sinx. (b) Plot several partial sums to illustrate the convergence of the Fourier series.If f(x) = 4,0= 1) The function f(x) periodic on the interval [0, 2л] has complex Fourier series f(x): Σ(1/n²) einx where the sum over n goes from - infinity to infinity. Convert this to cosine and sine Fourier Series by finding the values of A's and B's in the expression Ao + ΣAn cos(nx) + Σ Bn sin(nx) where each sum goes from 1 to infinity. Hint: consider the n and -n term together in the complex Fourier Series or use Euler's identity.+3 - nRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,