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C. Results from Analytic Function Theory

C.4 Functions of a Complex Variable

In the sequel, we will let $z=x+jy$denote a complex variable. Note that $z$ is not the argument in the Z-transform, as used at other points in the book. Also, a function $f(z)$ of a complex variable is equivalent to a function $\bar{f}(x,y)$. This will have real and imaginary parts $u(x,y)$and $v(x,y)$ respectively.

We can thus write

			\end{displaymath} (C.4.1)

Note that we also have

\begin{align*}\int_C f(z) dz&=\int_C (u(x,y)+jv(x,y))(dx+j dy)\\
		  &=\int_C u(x,y...
		  ...nt_Cv(x,y) dy
		  +j\left\{\int_C u(x,y) dy +\int_C v(x,y) dx \right\}

We then see that the previous results are immediately applicable to the real and imaginary parts of integrals of this type.