Complex variables

Please specify your notation(if necessary) and explain clearly each step of your solution. Thank you very much. 3. Sketch the region onto which the sector r ≤ 1, 0 ≤ _ ≤ π/4 is mapped by the transformation (a) _ = z2 (b) _ = z3 (c) _ = z4

Complex variable

Please specify your notation(if necessary) and explain clearly each step of your solution. Thank you very much. 7. Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation _ = ez , and label corresponding portions of the boundaries.

Complex Numbeers

Please see attached for question

Complex Variables : Limits

1. Prove the following (a), (b), and (c) by using this definition: for each positive number ε, there is a positive number δ such that | f (z) – ω0 | < ε whenever 0 < | z – z0 | < δ. (a) lim Re z = Re z0; z→z0 _ _ (b) lim z = ...continues

Complex Variables : Limits

3. Let n be a positive integer and let P(z) and Q(z) be polynomials, where Q(z0) ≠ 0. Find the following limits. (Please explain by using relevant theorem.) (a) lim 1/ zn (z0 ≠ 0) z→z0 (b) lim (iz3 -1) / (z + i) z→i (c) lim P(z) / Q(z) z→z0

Comples Variables : Limits

5. Show that the limit of the function _ f (z) = ( z / z )2 as z tends to 0 does not exist. Do this by letting nonzero points z = (x, 0) and z = (x, x) approach the origin. (Note that it is not sufficient to simply consider points z = (x, 0) and ...continues

Complex Variables : Limits

10. Show the following limits. (Please explain by using theorems.) (a) lim 4z2 / (z – 1)2 = 4 z→∞ (b) lim 1 / (z – 1)3 = ∞ z→1 (c) lim (z2 + 1) / (z – 1) = ∞ z→∞

Complex Variables : Limits

11. T (z) = (az + b) / (cz + d) (ad – bc ≠ 0) Show the following. (Please explain by using theorem.) (a) lim T (z) = ∞ if c = 0 z→∞ (b) lim T (z) = a / c and lim T (z) = ∞ if c ≠ 0. z→∞ z→-d/c

The problems are from complex variable class, 500 level in undergraduate.

The problems are from complex variable class. Please specify the terms that you use if necessary and explain each step of your solution. Thank you very much.

Complex Variables : Limits and Differentiability

3. Give a direct proof that f ΄(z) = -1 / z2 when f (z) = 1 / z (z ≠ 0). Use this definition to prove the problem. dw / dz = lim ∆w / ∆z ∆z→0