Holomorphic functions defined by integrals

Let $f$ be a continuous real valued function on $[0,1]$, and let the function $h$ in the complex plane be defined by

Prove that for fixed $\varphi$ the functions $g_{+}(\lambda ): = : = \left\langle (x+i 0)^{\lambda }, \varphi \right\rangle$ and $g_{-}(\lambda ): = \left\langle (x-i 0)^{\lambda }, \varphi \right\rangle$ are holomorphic on $\mathbb {C}$.

Define

What is the largest possible $P$ so that $h(z)$ is analytic for $\left|z\right| < P$?

Let $K \subset \mathbb {C}$ be a closed bounded set of positive (planar) measure. Define the function $\varphi$ by the equality

Prove that

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.

Define

Show that $f(z) = \bar{z}$ if $\left|z\right| < 1$ and that $f(z) = 1 / z$ if $\left|z\right| \geq 1$. Thus $f$ is not holomorphic in the unit disc, although the integrand is a holomorphic function of z. Note the contrast between this, on the one hand, and Theorem 10.7 and Exercise 16 on the other.

Suggestion: Compute the inner integral separately for $r < \left|z\right|$ and for $r > \left|z\right|$.

Let $E$ be a compact set on the real axis, of positive Lebesgue measure, let $\Omega$ be the complement of $E$, relative to the plane, and define

Answer the following questions:

Check your answers in Exercise 8 against the special case

Prove the regularity of the functions represented by the following integrals in the domains indicated in parentheses:

Let the function $\varphi (t)$ be continuous for $t \geqslant 0$ and satisfy the condition

Prove that the integral

represents a function regular in the half-plane $\mathrm{Re}\left(z\right) > \sigma$.

Let the function $\varphi (t)$ be continuous for $-\infty < t < \infty$ and satisfy the conditions

Prove that the integral

represents a function regular in the strip $\sigma_{1} < \mathrm{Im}\left(z\right) < \sigma_{2}$.

Find the domains of regularity of the functions represented by the following integrals:

Prove that the function $f(z) = \int_{0}^{\infty } \frac{e^{-t z}}{\sqrt{1+t^{2}}} d t$ is regular in the half-plane $\mathrm{Re}\left(z\right) > 0$ and that the point $z = 0$ is a singular point for it.

Let the function $\varphi (t)$, continuous and positive for $t \geqslant 0$, satisfy the condition

Prove that the function $f(z) = \int_{0}^{\infty } \varphi (t) e^{-t z} d t$ is regular in the half-plane $\mathrm{Re}\left(z\right) > 0$, and that the point $z = 0$ is a singular point for it.

Let the function $\varphi (t)$ be continuous for $t \geqslant 0$ and satisfy the condition

Prove that the function $f(z) = \int_{0}^{\infty } \varphi (t) e^{-t z} d t$ can be analytically continued to the half-plane $\mathrm{Re}\left(z\right) > -\delta$ with the point $z = i \alpha$ deleted, at which this function has a simple pole with principal part $\frac{1}{z-i \alpha }$.

Prove that the analytic continuation of each of the following functions has no singular points in the entire plane except poles. Find all the poles of these functions and the principal parts at these poles.

Let the function $\varphi (t)$ be continuous and positive on the segment $[0,1]$. Prove that the function $f(z) = \int_{0}^{1} \frac{\varphi (t)}{t-z} d t$, regular for $\mathrm{Im}\left(z\right) > 0$ (and even in the entire plane cut along the segment $[0,1]$), has singular points $z = 0$ and $z = 1$.

Let the function $\varphi (t)$ be regular in some domain containing the segment $[0,1]$. Prove that the function $f(z) = \int_{0}^{1} \frac{\varphi (t)}{t-z} d t$, regular for $\mathrm{Im}\left(z\right) > 0$, has singular points $z = 0$ and $z = 1$.

Suppose that $f(\omega , \cdot )$ is, for each $\omega$, a function on an open set $W$ in the complex plane and that $f(\cdot , z)$ is for $z$ in $W$ measurable $\mathscr {F}$ and integrable. Suppose that $A$ satisfies (16.9), that $f(\omega , \cdot )$ is analytic in $W$ for $\omega$ in $A$, and that for each $z_{0}$ in $W$ there is an integrable $g\left(\cdot , z_{0}\right)$ such that $\left|f^{\prime }(\omega , z)\right| \leq g\left(\omega , z_{0}\right)$ for all $\omega \in A$ and all $z$ in some neighborhood of $z_{0}$. Show that $\int f(\omega , z) \mu (d \omega )$ is analytic in $W$.

16.6 ↑ Show that a moment generating function $M(s)$ defined in $\left(-s_{0}, s_{0}\right)$, $s_{0}>0$, can be extended to a function analytic in the strip $\left[z:-s_{0}<\operatorname {Re} z<s_{0}\right]$. If $M(s)$ is defined in $\left[0, s_{0}\right), s_{0}>0$, show that it can be extended to a function continuous in $\left[z: 0 \leq \operatorname {Re} z<s_{0}\right]$ and analytic in $\left[z \cdot 0<\operatorname {Re} z<s_{0}\right]$.

Suppose $X$ is a random variable, and $f(z, X)$ is a jointly measurable function that is analytic in a neighborhood $\mathcal{N}: = \left\{ z \in \mathbb {C}:\left|z-z_{0}\right| < \delta \right\}$ of a point $z_{0}$ in the complex plane. Write $f^{\prime }(z, X)$ for the derivative with respect to $z$. Suppose $\left|f^{\prime }(z, X)\right| \leq M(X)$ for $\left|z-z_{0}\right| \leq \delta$, where $\mathbb {E}\left[M(X)\right] < \infty$. Show that $\mathbb {E}\left[f(z, X)\right]$ is analytic in $\mathcal{N}$, with derivative $\mathbb {E}\left[f^{\prime }(z, X)\right]$.