Convergence of complex integrals: Necessary and Sufficient conditions.

Question

Currently I am examining functions defined in the following way:

$F(z)=\int f(z,t) dt\ $where the integral is along some curve $\gamma\\$ not necessarily closed. I want to know necessary and sufficient for this integral to: exist, converge and be analytic in each case. I haven't been able to find much stuff online, if anyone could help me by providing some recommended texts or maybe explaining what the conditions are, with ideas for proof? Thanks

Answer 1

Suppose that, for each $z\in\mathbb C$, $f(t,z)$ is defined on some interval $(\alpha,\beta)$ (where $\alpha$ can be $-\infty$ and $\beta$ can be $+\infty$) and that:

1. the function $f\colon(\alpha,\beta)\times U\longrightarrow\mathbb C$ is continuous;
2. for each $z\in U$, the function$$\begin{array}{ccc}U&\longrightarrow&\mathbb C\\z&\mapsto&f(t,z)\end{array}$$is integrable and the integral $\int_\alpha^\beta f(t,z)\,\mathrm dt$ converges uniformly on every compact subset of $U$.

Then the function$$\begin{array}{rccc}f\colon&U&\longrightarrow&\mathbb C\\&z&\mapsto&\displaystyle\int_\alpha^\beta f(t,z)\,\mathrm dt\end{array}$$is analytic and$$f'(z)=\int_\alpha^\beta \frac{\partial f}{\partial z}(t,z)\,\mathrm dt.$$

The main theorem to prove this is a theorem due to Weierstrass: if $(f_n)_{n\in\mathbb N}$ is a sequence of analytic function from $U$ into $\mathbb C$ which converges uniformly to $f\colon U\longrightarrow\mathbb C$, then $f$ is analytic and $(f_n')_{n\in\mathbb N}$ converges uniformly on every compact of $U$ to $f'$.