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- Write the Laurent series expansion in powers of z – 1 for 1 f(2) = which is valid in z – 1| > 2. (z – 1)(2² – 1)’Need help with this Intro to Elementary Number Theory Homework problem.1.) Find the Fourier series for the function f(t) = (1,-4 < t ≤0 2,0Q.3-b-E dt² Q.4-Expand Fourier series in cosines only defined by: (15 f(x) = π-x, 01. Find the Fourier series expansion of the following periodic function: -2 < t <0 02. Find the Fourier series for the function f(x) = |x|, -T < x < T; f(x+ 27) = f(x). Use this series to find the sum for the infinite series E . n=1 n2:?Find the Taylor's or Laurent's series expansion of the complex variable function which is represented by : f(z) = ; i) 1< ]z] < 2 ii) ]z|< 2. Also (z²–1)(z²+4) z2 classify the singularity of f(z) %3D (z-2)ez-1Expand the following periodic function into a Fourier series. for -2 < <-T < 0 -2 A f(x) = 1 0 for T-X for X-T for k <-T 0 < ㅠ XX1. Find the Fourier series expansion of the following periodic function: -2 < t <0 0- A Fourier Series a.) Determine the Fourier series expansion for the periodic extension of (x, ß) cosh(ßx) for -n < x < T, where B is a fixed positive constant. b.) Then use this to compute exact values for the sums (-1)" + B2 1 Σ Σ and n? n=1 + B2 n=1 as functions of ß for 0 < B. Hint: Try evaulating f(x,ß) at different values of x.3. Find the fourier series expansion of the periodic function f(r) = e², -a < r < 7, 1 1 f(1+ 2n) = f(r). Hence obtain the sum of the series 1 1+22 1+32'1+ 4? (-1)" +.... 1+n?SEE MORE QUESTIONSRecommended 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,