(a) Prove that Σ³ = n²(n+1) ². j=1 (b) Evaluate fr³ de directly from the definition, without using the Fundamen- tal Theorem of the Calculus. (Hint: use Lemma 4.13 or Theorem 4.16.)

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A: Q.1 If fi(z) and f2(z) are both analytic functions, show that f3(z)=f1(z).f2(z) is also analytic…

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A:

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Q: Q.1 If fi(z) and fa(z) are both analytic functions, show that fa(z)-fi(z).f2(z) is also analytic…

A:

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A: We need to find the inverse function of the real valued function f(x)=2x+9

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Q: Use the Residue theorem to evaluate the integral fe(z-2)dz.

A: Use the expansion of e1z  =1 + 1z + 12! z2 + 13! z3 + ...  Hence ∫e1z(z-2)dz = ∫(z-2)1 + 1z + 12! z2…

Q: -) Consider the function f: R\{-1} → R given by f(z): your claim. z+1' Find the image off and prove

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Q: 12. Evaluate the expression using de Moivre's Theorem. Write your answer in standard form z = x +…

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Q: 3. Express the following in terms of z] and Zz: (a) z* (b) 1/z (c) Re{z} (d) Im{z}

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A: The given function

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Q: 1. Find an antiderivative of f(x) = 16–x². Then, use The Fundamental Theorem of Calculus %3D 4 to…

A: Given : fx = 16-x2

Q: 5. Suppose that f: R→ R is continuous. and f(x) > 0.. Prove that there is an in- terval I =…

A: Used the definition of continuity.