3.2 參考解答

1. (b) $f(z)=f(x,y)=\frac{y+ix}{x^2+y^2}$

   Let $u(x,y)=\frac{y}{x^2+y^2}$ , $v(x,y)=\frac{x}{x^2+y^2}$

   $u_x=\frac{-y \cdot (2x)}{(x^2+y^2)^2}$ , $u_y=\frac{(x^2+y^2)-y \cdot(2y)}{(x^2+y^2)^2}=\frac{x^2-y^2}{(x^2+y^2)^2}$

   $v_x=\frac{(x^2+y^2)-x \cdot (2x)}{(x^2+y^2)^2}=\frac{y^2-x^2}{(x^2+y^2)^2}$ , $v_y=\frac{-x \cdot (2y)}{(x^2+y^2)^2}$

   satisfy $\begin{cases} u_x=v_y \\ u_y=-v_x \end{cases}$

   Hence, $f’(z)=u_x+iv_x=\frac{-2xy+i(y^2-x^2)}{(x^2+y^2)^2}$, $\forall \space (x,y) \in \mathbb{C}$ except $(0,0)$.

   (c) $f(z)=-2(xy+x)+i(x^2-2y-y^2)$

   Let $u(x,y)=-2xy-2x$ , $v(x,y)=x^2-2y-y^2$

   $u_x=-2y-2$ , $u_y=-2x$

   $v_x=2x$ , $v_y=-2-2y$

   satisfy $\begin{cases} u_x=v_y \\ u_y=-v_x \end{cases}$

   Hence, $f’(z)=(-2y-2)+i(2x)$, $\forall \space (x,y) \in \mathbb{C}$.

   (h) $f(z)=|z-(2+i)|^2=|(x+iy)-(2+i)|^2=(x-2)^2+(y-1)^2$

   Let $u(x,y)=(x-2)^2+(y-1)^2$ , $v(x,y)=0$

   $u_x=2(x-2)$ , $u_y=2(y-1)$ , $v_x=v_y=0$

   satisfy $\begin{cases} u_x=2(x-2)=0=v_y \\ u_y=2(y-1)=0=-v_x \end{cases}$

   $\Rightarrow (x,y)=(2,1)$

   Hence, $f’(z)$ is differentiable at $(2,1)$ & $f’(2+i)=0$.

3. $f(z)=(2x-y)+i(ax+by)$

5. $f(z)=e^x \cos{y}+ie^x \sin{y}$

   Let $u(x,y)=e^x \cos{y}$ , $v(x,y)=e^x \sin{y}$

   $u_x=e^x \cos{y}$ , $u_y=-e^x \sin{y}$

   $v_x=e^x \sin{y}$ , $v_y=e^x \cos{y}$

   $1^。$ $u_x$, $u_y$, $v_x$, $v_y$ are continuous, $\forall \space (x,y) \in \mathbb{C}$

   $2^。$ $u_x=v_y$, $u_y=-v_x$

   By CRT, $f’(z)=e^x \cos{y}+ie^x \sin{y}$ & $f(z)$ is analytic, then $f$ is differentiable.

   Similary, $f’(z)$ is differentiable & $f’’(z)=e^x \cos{y}+ie^x \sin{y}$.

   Note, $f(z)=e^z=e^{x+iy}=e^x \cdot e^{iy}=e^x(e^x \cos{y}+ie^x \sin{y})$,

   $f’(z)=e^z$, $f’’(z)=e^z$

8. (a) $f(z)=f(re^{i\theta})=\ln{r}+i\theta$, $r>0$, $-\pi<\theta<\pi$

9. (a) $f(z)=\cosh{x}\sin{y}-i\sinh{x}\cos{y}$

   Let $u(x,y)=\cosh{x}\sin{y}$, $v(x,y)=-\sinh{x}\cos{y}$

   $u_x=\sinh{x}\sin{y}$, $u_y=\cosh{x}\cos{y}$

   $v_x=-\cosh{x}\cos{y}$, $v_y=\sinh{x}\sin{y}$

   satisfy $u_x=v_y$, $u_y=-v_x$ & $u_x$, $u_y$, $v_x$, $v_y$ are continuous, $\forall \space (x,y) \in \mathbb{C}$

   Then $f(z)$ is analytic function & $f’(z)=\sinh{x}\sin{y}-i\cosh{x}\cos{y}$.

   Hence, $f(z)$ is entire.

11. (b) $f(z)=8x-x^3-xy^2+i(x^2y+y^3-8y)$