MAT334-2018F > Quiz-5

Q5 TUT 0202

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Victor Ivrii:
$\newcommand{\Log}{\operatorname{Log}}$

Use Morera's Theorem to show that the following function is analytic on the indicated domain; find a power-series expansion for the function by using the known power series for the integrand and interchanging the summation and integration.
$$\int_0^{1/2} \Log (1-tz)\,dt\qquad\text{on}\; |z|<2.$$

ZhenDi Pan:
We have
\begin{equation}
\int_{0}^{1/2} \log(1 - t z) dt
\end{equation}
Integrate over $\gamma$ with respect to $z$, consider
\begin{equation}
f(z)=\log(1-tz)
\end{equation}
Function $f(z)$ is analytic on $\mid z \mid <2$, by Cauchy's theorem, for any closed curve $\gamma$
\begin{equation}
\int_\gamma f(z)dz = 0 \\
\int_{0}^{1/2} (\int_\gamma \log(1-zt) \,dz)\,dt = \int_{0}^{1/2} 0\,dt
\end{equation}
So it is analytic on domain $\mid z\mid < 2$.

Since $\log(1 - t z) = \sum_{n=1}^\infty \frac{-( z t)^n}{n}$ is valid when $\mid zt \mid<1$, and since $\mid z \mid<2$, for all  $t \in [0,\frac{1}{2}]$. We have
\begin{equation}
\int_{0}^{1/2} \log(1 - t z) dt  = - \int_{0}^{1/2} \sum_{n=1}^\infty  \frac{( z t)^n}{n} dt \\
= - \sum_{n=1}^\infty  \int_{0}^{1/2}  \frac{( z t)^n}{n} dt \\
=  -\sum_{n=1}^\infty\frac{1}{2^{n+1} n (n+1)} z^n
\end{equation}

Victor Ivrii:
It is $\Log$ rather than $\log$ ; otherwise it would be multivalued