Morera's theorem

Prove that the uniform limit of a sequence of complex analytic functions is complex analytic. Is the analogous theorem true for real analytic functions?

Let $f$ be a continuous complex valued function on $[0,1]$, and define the function $g$ by

Prove that $g$ is analytic in the entire complex plane.

Let $f$ be continuous on $\mathbb {C}$ and analytic on $\left\{ z \mid \mathrm{Im}\left(z\right) \neq 0\right\}$. Prove that $f$ must be analytic on $\mathbb {C}$.

Define the function $f$ on $\mathbb {C} \setminus [0,1]$ by

Prove that $f$ is analytic, and find its Laurent series about $\infty$.

The complex function $f(z)=u(x,y)+iv(x,y)$ of the real variables $x$ and $y$ is defined and continuous in a region $\mathfrak {R}$ of the $z=x+iy$ plane. Moreover we assume that the integral

along any circle inside $\mathfrak {R}$ vanishes. Then $f(z)$ is an analytic function of the complex variable $z$, regular in the entire region $\mathfrak {R}$.

Suppose $\mu$ is a complex measure on a measure space $X, \Omega$ is an open set in the plane, $\varphi$ is a bounded function on $\Omega \times X$ such that $\varphi (z, t)$ is a measurable function of $t$, for each $z \in \Omega$, and $\varphi (z, t)$ is holomorphic in $\Omega$, for each $t \in X$. Define

for $z \in \Omega$. Prove that $f \in H(\Omega )$. Hint: Show that to every compact $K \subset \Omega$ there corresponds a constant $M < \infty$ such that

Determine the regions in which the following functions are defined and holomorphic:

Hint: Either use Exercise 16, or combine Morera's theorem with Fubini's.

Suppose that $I = [a, b], \Omega$ is a region, $I \subset \Omega , f$ is continuous in $\Omega$, and $f \in H(\Omega -I)$. Prove that actually $f \in H(\Omega )$.

Replace $I$ by some other sets for which the same conclusion can be drawn.