Rachel

Hi class

JING_Picture_3-14-14.png

Question 2

Find the real and imaginary parts in terms of rectangular coordinates x and y.

(1)
\begin{eqnarray} a) w & = & 6z+5\\ & = & 6(x+iy)+5\\ & = & 6x+6iy+5\\ &&\\ &&u=6x+5\\ &&v=6y\\ \end{eqnarray}
(2)
\begin{eqnarray} b) w & = & z^3-2z\\ & = & (x+iy)^3-2(x+iy)\\ & = & (x+iy)(x+iy)(x+iy)-2(x+iy)\\ & = & (x^2+2xiy+i^2y^2)(x+iy)-2(x+iy)\\ & = & (x^3+2x^2iy+i^2y^2x+x^2iy+2xi^2y^2+i^3y^3)-2(x+iy)\\ & = & (x^3+3x^2iy+3xi^2y^2+i^3y^3)-2(x+iy)\\ & = & (x^3+3x^2iy-3xy^2-iy^3)-2(x+iy)\\ & = & (x^3+3x^2iy-3xy^2-iy^3-2x-2iy\\ &&\\ &&u=x^3-3xy^2-2x\\ &&v=3x^2y-y^3-2y\\ \end{eqnarray}
(3)
\begin{eqnarray} c) w & = & \frac{1}{z}\rightarrow \frac{1}{x+iy} \rightarrow \frac{(1)(x-iy)}{(x+iy)(x-iy)}\\ & = & \frac{x-iy}{x^2-xiy+xiy-i^2y^2}\\ & = & \frac{x-iy}{x^2-i^2y^2} \rightarrow \frac{x-iy}{x^2+y^2}\\ & = & \frac{x}{x^2+y^2}- \frac{iy}{x^2+y^2}\\ &&\\ &&u= \frac{x}{x^2+y^2}\\ &&v= -\frac{y}{x^2+y^2}\\ \end{eqnarray}
(4)
\begin{eqnarray} d) w & = & e^z\\ & = & e^{x+iy}\\ & = & e^xe^{iy}\\ & = & e^x(cosy+isiny)\\ \end{eqnarray}
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