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Lebesgue’s dominated convergence theorem extends the idea of interchanging limits and integrals to lim fn(c)dx = lim fn(r)dz, with a, b âˆˆ â„ \ {âˆž} and n âˆˆ â„• and fn converges point-wise (meaning for each fixed x âˆˆ (a,b) the sequence {fn(x)} is convergent). Apply this theorem and Theorem 24.1 of Lecture 24. You do not need to prove the theorems nor do an Îµ – N argument. However, do not forget to check the requirements for these theorems! We assume -âˆž < a < b < âˆž real and finite and Q(2) rational function with poles at {2c}kL,L [a, b].

Before you compute the two integrals below, explain why it is f(z) f(z) lim Res Res nâ†’âˆž 2z0 0n 2Fz0

for any function holomorphic in punctured neighborhood of z0 and the convergent sequence lim nâ†’âˆž an = a with a, an = 2 for all n âˆˆ â„•. (Hint: think of the contour integral representation of the residue and Lebesgue’s dominated convergence theorem:

Calculate the integral Q(âˆž) dx with Q(2) = O(2^-2) when 2 = 0 by expressing it as a finite sum of residues. (Hint: consider the limit lim Vâ†’0 âˆ‘(n-1)

Calculate the integral âˆ® dz with Q(2) = O(z^-1) when 2 â‰ 0 by expressing it as a finite sum of residues. (Hint: see the hint of part (b).)

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01:59

Let (ancos(nx) + bsin(nx)) be the Fourier series of f in the interval [-T,T] in R, and let (2a, cos(nx) + b’nsin(x)) be the Fourier series of the derivative of f. In the proof of the Uniform Convergence Theorem, we show that the Fourier series of f converges absolutely and uniformly. In particular, we show that both series Î£ |an| and Î£ |bn| converge. We then apply the Weierstrass M-test with Mn = |an| + |bn| to show that the series Î£ (an*cos(nx) + bn*sin(nx)) converges. We also show that the series Î£ ((an)^2 + (bn)^2) converges by applying Bessel’s Inequality. We use integration by parts and the assumption that f(-T) = f(T) to show that the series Î£ (a’n*cos(nx) + b’n*sin(nx)) converges, since both series Î£ (a’n)^2 and Î£ (b’n)^2 converge for n = 1,2,3,… The sequence {Sn} of the partial sums of the series Î£ |an| converges, since Î£ |an| is bounded by applying the Schwartz inequality.

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