ToTopics will be chosen from the following: Elliptic equations in two independent variables: Harmonic functions. Mean value property. Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of solutions in terms of Green’s functions.

这是一份Bath巴斯大学MA30059 作业代写的成功案

$$
|\Gamma(z)| \leq \Gamma(x), \quad \operatorname{Re}(z)=x>0
$$
If $f(z)$ is a complex function of $z$, then writing $f(z)$ as
$$
f(z)=u(x, y)+i v(x, y),
$$
it follows that the Cauchy-Riemann equations [7] must be satisfied, i.e.,
$$
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}
$$
For $f(z)=\Gamma(z)$, we have, using the relation,
$$
\begin{aligned}
t^{z} &=e^{z \ln t}=e^{x \ln t+i y \ln t} \
&=e^{x \ln t}[\cos (y \ln t)+i \sin (y \ln t)]
\end{aligned}
$$

MA30059 COURSE NOTES ：

Therefore, we conclude that
$$
R(m, n) \equiv \int_{0}^{1} x^{m}(\ln x)^{n} d x=\frac{(-1)^{n} n !}{(m+1)^{n+1}}
$$
Note that for $n=0$, we obtain
$$
R(m, n)=\int_{0}^{1} x^{m} d x=\frac{1}{m+1}
$$
and for $m=0$,
$$
R(0, n)=\int_{0}^{1}(\ln x)^{n} d x=(-1)^{n} n !
$$