### Complex Variables 2.2.17

Find the limit of f(z) = x^2/(x^2+y^2) +2i Where z=x+iy and |z| --> 0

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Find the limit of f(z) = x^2/(x^2+y^2) +2i Where z=x+iy and |z| --> 0

All steps must be shown, even small details. Workout the relevant calculations please complete the problem, not just set it up. Provide common sense explanations. Bonus: The function p(z)=[z(z+1)]^-1 can be written in two different ways: (see attached for full equation) These two expansions are contradictory. The first

All steps must be shown, even small details. Workout the relevant calculations please complete the problem, not just set it up. Provide common sense explanations. 6. For a>0, use residue calculus to evaluate (see the attachment for the full problem description and equation.)

Compute: lim as r -> infinity |f(z)|, where z = re^[(i)("alpha")] Your answer will depend on "alpha". Hint: First consider what curve z = re^[(i)("alpha")] traces out in the complex plane as r -> infinity

Please see the attached file for full problem description.

What are all possible solutions of z^4 + 4 = 0? From this information, write out a complete factorization of z^4 + 4.

Please see attachments

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Due to the amount of symbols in the problem I attached a .pdf file. In words, I think the problem is asking - if the limit of f at c is positive, then on some deleted neighborhood of c, f is positive away from 0.

Please see the attached file for the fully formatted problem. For phi E C2[R3 ! R3], curl grad phi = 0. Prove this. The converse is "Poincare's Lemma": if f E C1[R3 --> R3] and if curl f = 0, then f is a gradient, i.e., f = grad for some 2 C2. Try it this way: if f = grad phi, then phi (x1, x2, x3) = phi(0)+ ....

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Please see the attached file for the fully formatted problems. Consider C[0, 1], the space of real valued continuous functions defined on the unit interval [0, 1]. Let K = C1[0, 1] {f : Z 1 0 f02 1, ||f||1 1} Note that C1[0, 1] C[0, 1], and K C[0, 1]. Show that K is compact in C. I am assuming compactness h

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