# Toronto Math Forum

## MAT334-2018F => MAT334--Tests => Quiz-5 => Topic started by: Victor Ivrii on November 02, 2018, 03:30:30 PM

Title: Q5 TUT 0202
Post by: Victor Ivrii on November 02, 2018, 03:30:30 PM
$\newcommand{\Log}{\operatorname{Log}}$

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.
$$\int_0^{1/2} \Log (1-tz)\,dt\qquad\text{on}\; |z|<2.$$
Title: Re: Q5 TUT 0202
Post by: ZhenDi Pan on November 02, 2018, 05:06:37 PM
We have

\int_{0}^{1/2} \log(1 - t z) dt

Integrate over $\gamma$ with respect to $z$, consider

f(z)=\log(1-tz)

Function $f(z)$ is analytic on $\mid z \mid <2$, by Cauchy's theorem, for any closed curve $\gamma$

\int_\gamma f(z)dz = 0 \\
\int_{0}^{1/2} (\int_\gamma \log(1-zt) \,dz)\,dt = \int_{0}^{1/2} 0\,dt

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

\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
Title: Re: Q5 TUT 0202
Post by: Victor Ivrii on November 04, 2018, 09:47:04 PM
It is $\Log$ rather than $\log$ ; otherwise it would be multivalued