Help me fast so that I will give good rating. 1- z1. For the function f(z) =evaluate f(2i) and f(-1+i), describe its domain, and write27 +i'it in the form of u(x, y) + iv(x + y).32. Determine the values of z for which the function f(z) =is undefined.%3D(24 + 8iz)3. Express the function a? + ayi in terms of z and z.4. Express the functionf(2) = iz + Im(i/z), z # 0in the form of u(x, y) + iv(x + y).5. Determine the points that are not in the domain of the functionf(2) =z4 + 2/3+ 2i6. Evaluate the following limits.(a) lim;»i(z² – 27)(b) lim;→1+2i[x? + y² + i cos(xy/2)]7. Let g(z) = (2² + 9)/(z – 3i). Is g continuous at the point z = 3i? If not, redefine g suchthat it is continuous at the point?8. Is the following function continuous at the point z + i?22 +1if z + if(2) =3i,2 - 2if z = i9. Determine whether the function22 + 4if z + -2ig(z) =2(z+ 2i)'0,if z = -2iis continuous at the point z = -2i.

Answered: 1- z 27 + i' it in the form of u(x, y)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Help me fast so that I will give good rating.

1- z
1. For the function f(z) =
evaluate f(2i) and f(-1+i), describe its domain, and write
27 +i'
it in the form of u(x, y) + iv(x + y).
3
2. Determine the values of z for which the function f(z) =
is undefined.
%3D
(24 + 8iz)
3. Express the function a? + ayi in terms of z and z.
4. Express the function
f(2) = iz + Im(i/z), z # 0
in the form of u(x, y) + iv(x + y).
5. Determine the points that are not in the domain of the function
f(2) =
z4 + 2/3+ 2i
6. Evaluate the following limits.
(a) lim;»i(z² – 27)
(b) lim;→1+2i[x? + y² + i cos(xy/2)]
7. Let g(z) = (2² + 9)/(z – 3i). Is g continuous at the point z = 3i? If not, redefine g such
that it is continuous at the point?
8. Is the following function continuous at the point z + i?
22 +1
if z + i
f(2) =
3i,
2 - 2
if z = i
9. Determine whether the function
22 + 4
if z + -2i
g(z) =
2(z+ 2i)'
0,
if z = -2i
is continuous at the point z = -2i.

Expert Solution

Step 1

Note: As per bartleby instruction when more than one question is given only one has to be answered. Please upload other separately.

Given $f (z) = \frac{1 - z}{2 z + i}$

The value for $f (2 i)$ is,

$\begin{array}{rcl} f (2 i) & = & \frac{1 - 2 i}{2 (- 2 i) + i} \\ = & \frac{1 - 2 i}{- 4 i + i} \\ = & \frac{1 - 2 i}{- 3 i} \\ = & \frac{2 i^{2} - i}{- 3} \\ = & \frac{2 + i}{3} \end{array}$

Step 2

The value for $f (- 1 + i)$

$\begin{array}{rcl} f (- 1 + i) & = & \frac{1 - (- 1 + i)}{2 (- 1 - i) + i} \\ = & \frac{2 - i}{- 2 - 2 i + i} \\ = & \frac{2 - i}{- 2 - i} \\ = & \frac{2 - i}{- 2 - i} \times \frac{2 - i}{2 - i} \\ = & \frac{4 - 1 - 4 i}{- 4 + 1} \\ = & \frac{3 - 4 i}{- 3} \\ = & \frac{4 i - 3}{3} \end{array}$

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