1. 求下列各式的微分

（1）$1+e^z$

f = 1 + E^z
D[f, z]

（2）$\sin z+\cos z$

f = Sin[z] + Cos[z]
D[f, z]

2. 函数$f(z)=z|z{|}^2$在何处可导？何处解析？

Clear[f]
Clear[z]
$Assumptions = x \[Element] Reals
$Assumptions = y \[Element] Reals
z = x + I*y;
f = z*(Abs[z])^2;
u = ComplexExpand[Re[f]]
v = ComplexExpand[Im[f]]
dux = D[u, x]
duy = D[u, y]
dvx = D[v, x]
dvy = D[v, y]
sol = Solve[dux == dvy && duy + dvx == 0, {x, y}]

3. 求解下列方程

（1）$\ln z=\frac{\pi i}{2}$

Clear[z];
sol1 = Solve[Log10[z] - I*Pi / 2 == 0, z]
Reduce[Log10[z] - I*Pi / 2 == 0, z]

（2）$1+e^z=0$

Clear[z];
sol2 = Reduce[1 + E^z == 0, z]

（3）$e^z-1-\sqrt{3}i=0$

Clear[z];
sol3 = Reduce[E^z - 1 - I*Sqrt[3] == 0, z]

4. 设f(z)为解析函数，求系数a,b,c,d的值

$f(z)=a{y}^3+b{x}^{2}y+i(x^3+cx{y}^2)$为解析函数

Clear[x];
Clear[y];
Clear[u];
Clear[v];
u[x_, y_] := a*y^3 + b*x^2*y;
v[x_, y_] := x^3 + c*x*y^2;
dux = D[u[x, y], x]
duy = D[u[x, y], y]
dvx = D[v[x, y], x]
dvy = D[v[x, y], y]
Solve[2*b - 2*c == 0 && b + 3 == 0 && 3 a + c == 0, {a, b, c}]

5. 计算积分

（1）${\int }_{|z|=1}\frac{dz}{z}$
方法一：积分法

Integrate[I*E^(I*theta) / E^(I*theta), {theta, 0, 2*Pi}]

方法二：留数求解法

Residue[1 / z, {z, 0}]*2*Pi*I

（2）${\int }_{|z|=1}\frac{dz}{|z|}$
方法一：积分法

Integrate[I*E^(I*theta) / Abs[E^(I*theta)], {theta, 0, 2*Pi}]

方法二：留数求解法

Residue[1 / Norm[z], {z, 0}]*2*Pi*I

6. 求隐函数的导数或偏导数

（1）$\ln \sqrt{x^2+y^2}=\arctan \frac{y}{x}$

Clear[x];
Clear[y];
Clear[u];
Clear[v];
D[Log10[Sqrt[x^2 + y^2]] - ArcTan[y/x] == 0, y, NonConstants -> x]
% /. D[x, y, NonConstants -> {x}] -> x'
Solve[%, x']

（2）$x=u\cos \frac{v}{u},y=u\sin \frac{v}{u}$，求$\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}$

F1 = x1 - u1*Cos[v1/u1];
F2 = y1 - u1*Sin[v1/u1];
F1x = D[F1, x1]
F1u = D[F1, u1]
F1y = D[F1, y1]
F1v = D[F1, v1]
F2x = D[F2, x1]
F2y = D[F2, y1]
F2u = D[F2, u1]
F2v = D[F2, v1]
u2x = -(F1x*F2v - F1v*F2x)/(F1u*F2v - F1v*F2u)
u2y = -(F1y*F2v - F1v*F2y)/(F1u*F2v - F1v*F2u)
v2x = -(F1u*F2x - F1x*F2u)/(F1u*F2v - F1v*F2u)
v2y = -(F1u*F2y - F1y*F2u)/(F1u*F2v - F1v*F2u)