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- Let g(t) have period T and be defined as g(t) = (t+ 2) for -Test the given integrals for convergence. Choose two (2) items only. 4+ cos(x) dx (a) So ∞ 1 + cos² (1) da √1+x² (b) √o (1+2) √=" 0 (c) √ ₁ 2²- ra-e-dr, a>03a) Find the circle of convergence for the complex power series (2-1-1)" show it (draw it) on the complex plane. b) Verify that cosh(z) = cosh(x) cos(y) + i sinh(r) sin(y) anda. determine the Fourier coefficients. b. Write the corresponding series. c. Expand the Fourier series up to at least n = 5. d. Write the convergence of the series e. Use a graphing program to check that the Fourier series converges to the indicated functionQ3. a) Evaluate the integral: [ (x² + y²) dx dy b) Find Laplace transform of the function: f(t)=e" (t.−1)². (x + 2)" Q4. Find the convergence interval and radius of the series : (-1)**!! n 1Determine the nth partial sum of the Fourier Series of: + x, - T < x < 0 f(x) = х, 2 0Q) find Fourier series on [-7,1] – 1 |4 -π-4h (a) Derive the Gaussian quadrature formula of the form fh f(x) dx = a f(h) + bf (2h) + cf (3h). Find the order of convergence of this formula. (b) Apply the formula you found on part (a) to X₁+k = x₁ + fti+k f(t, x(t))dt, to find an approximate formula for the numerical solution of the IVP (t) = f(t, x(t)), x(t₁) = xo. (c) Use the IVP (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the interval [0, 2] with h = 0.25. [Hint: the exact solution is x(t) = (2) t²-t.-1, -n sx S What is the coefficient value of (a,) for Fourier Series of the function f(x) = -< x s 0, 1, b) 3 d) f) o g) None of themRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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