Riemann surface
matlab

Consider a force field $\vec{F}(x, y)=-y \vec{i}+x \vec{j}$ in the plane and a curve $\mathcal{C}$ given by $x(t)=a \cos (t)$ and $y(t)=b \sin (t)$ with $0 \leq t \leq \pi$ and positive constants $a$ and $b$.

Compute $\int_{\mathcal{C}} \vec{F} \cdot d \vec{r}$ using the definition of the line integral. Does $\vec{F}$ have a potential?

Let a different force field be given by $\vec{G}=2 x y \vec{i}+\left(x^2-y^2\right) \vec{j}$. Does $\vec{G}$ have a potential? If yes, find the potential of $\vec{G}$.

Calculate $W=\int_\gamma \vec{G} \cdot d \vec{r}$ where $\gamma$ is the semi-circle connecting the points $(0,-1)$ and $(0,1)$. Use again the definition of the line integral.

Calculate $W$ again by using a line integral connecting the above points. Can you think of a third way to get $W$ easily?

Find the Laurent series for the following function
$$f(z)=\frac{1}{z^2-(2+i) z+2 i}$$

The complex Bessel function $J_n(z)$ of order $n$ can be defined for an integer $n$ by
$$\mathrm{e}^{z(w-1 / w) / 2}=\sum_{n=-\infty}^{\infty} J_n(z) w^n$$
Hence, $J_n(z)$ is the coefficient of $w_n$ in the Laurent series of $\exp (z(w-1 / w) / 2)$, as a function of $w$, about 0 .
(a) Use the integral formula for Laurent coefficients to show that
$$J_n(z)=\frac{1}{\pi} \int_0^\pi \cos (n \theta-z \sin \theta) d \theta$$
(b) Write
$$\mathrm{e}^{z(w-1 / w) / 2}=\mathrm{e}^{z w / 2} \mathrm{e}^{-z /(2 w)}$$
and multiply the series expansions of these function about 0 to obtain
$$J_n(z)=\sum_{k=0}^{\infty}(-1)^k \frac{1}{k !(n+k) !}\left(\frac{z}{2}\right)^{n+2 k}$$

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