User:Selfworm/Math2: Difference between revisions

We are interested in studying how complex analytic functions behave when limited to the real line. So, let $\mathbb {D}$ be a domain that contains some part of the real line and let $f(x)$ be a complex analytic function from a domain$\mathbb {D}$ into $\mathbb {C}$. Since $f$ is analytic it has harmonic conjugates, call them $u(x,y)$ and $v(x,y)$. So,

$f(z)=u(x,y)+iv(x,y)$

An important result follows:

$f'(z)$	$=\int f'(z)\,dz$
	$=\int u_{x}(z,0)+iu_{y}(z,0)\,dz$
	$=\int v_{y}(z,0)+iv_{x}(z,0)\,dz$

We will derive the above result by using the conjugates of $f'(x)$. The domain of f and ${\overline {f}}$ need not be the same so we will let ${\overline {\mathbb {D} }}=\{z:{\overline {z}}\in \mathbb {D} \}$, and $\mathbb {D^{*}} =\mathbb {D} \bigcap {\overline {\mathbb {D} }}$. In this way both $f$ and ${\overline {f}}$ will be defined on $\mathbb {D^{*}}$. Now since $f$ is analytic on $\mathbb {D^{*}}$ we have that

$f'(z)$	$=u_{x}(x,y)+iu_{y}(x,y)$
	$=v_{y}(x,y)+iv_{x}(x,y)$

By the substitution y = 0 and the Cauchy-Riemann equations we arrive at

$f'(z)$	$=u_{x}(x,y)+iu_{y}(x,y)$
	$=v_{y}(x,y)+iv_{x}(x,y)$

To simplify notation we let $F=f'$ and we note that,

${\overline {F(z)}}=u_{x}(x,-y)+iu_{y}(x,-y)$