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## Question 1 : Part (i)

Evaluate $(1+3i)+(5+7i)$

Add the real parts and add the imaginary parts:

$(1+5)+(3+7)i$

Simplify:

$6+10i$

**Express Geometrically:** When you add 2 complex numbers, you can think of vector addition.

The Answer is the diagonal of the parallelogram.

## Question 1B

Evaluate the following contour integrals by parameterizing the contouring:

$\int _{c} \frac{z+2}{z}dz\; \;,$ where C is the upper half of circle of radius 2 centered at origin

$z=z0+re^{iθ}, θ->[0,pi]$

$z=2e^{it}, t->[0,pi]$

$dz=2ie^{it}dt$

$\int_{0}^{pi} \frac{2e^{it}+2}{2e^{it}}\, 2ie^{it} dt$

$\int_{0}^{pi}\frac{2ei^{2it}=2ie^{it}}{e^{it}} dt$

$\int_{0}^{pi} 2ie^{it} + 2i dt$

$2\int_{0}^{pi} ie^{it}+2i dt$

$2[e^{it}+2it]$ from 0 to pi

$2[(e^{ipi+2ipi})-(e^{0}+0)]$

$2[cos(pi)+isin(pi)+2ipi-1]$

$2[2ipi-2]$

$4ipi-4$

++Question 1: Part (ii)

Evaluate $(1+3i)-(5+7i)$

Distribute the negative:

$1+3i-5-7i$

Add real and imaginary parts:

$(1-5)+(3i-7i)$

Simplify:

$-4-4i$

**Express Geometrically:** When you subtract 2 complex numbers, you distribute the negative then think of vector addition.

++Question 7

Find the image of the unit square 0<x<1, 0<y<1, under mapping w=Z^{2}.

Provide some algebra.