这是很早之前貌似从一本书上学习 Mathematica 时写的，算法原理也是书上的，年代太久远，我记不清楚出处了。

定义两个函数，分别是 CartesianMap 和 PolarMap 用来处理直角坐标和极坐标下的绘图，可接受绘图参数来控制图形显示效果。(这两个函数后来加入系统扩展包中了，再到后来又取消了，见评论链接，不过仍然不失为一种不错的方法)具体用法我举几个例子说明一下。

$Lines = 15;

Options[CartesianMap] = Options[PolarMap] = {Lines -> $Lines};

CartesianMap[

func_, {x0_, x1_, dx_: Automatic}, {y0_, y1_, dy_: Automatic},

opts___?OptionQ] /;

VectorQ[{x0, x1, y0, y1},

NumericQ] && (NumericQ[dx] ||

dx === Automatic) && (NumericQ[dy] || dy === Automatic) :=

Module[{x, y},

Picture[CartesianMap,

func[x + I*y], {x, x0, x1, dx}, {y, y0, y1, dy}, opts]];

PolarMap[func_, {r0_: 0, r1_, dr_: Automatic}, {p0_, p1_,

dp_: Automatic}, opts___?OptionQ] /;

VectorQ[{r0, r1, p0, p1},

NumericQ] && (NumericQ[dr] ||

dr === Automatic) && (NumericQ[dp] || dp === Automatic) :=

Module[{r, p},

Picture[PolarMap,

func[r*Exp[I*p]], {r, r0, r1, dr}, {p, p0, p1, dp}, opts]];

PolarMap[func_, rr_List, opts_?OptionQ] :=

PolarMap[func, rr, {0, 2*Pi}, opts];

Picture[cmd_, e_, {s_, s0_, s1_, ds_}, {t_, t0_, t1_, dt_}, opts___] :=

Module[{hg, vg, lines, nds = ds, ndt = dt},

lines = Lines /. {opts} /. Options[cmd];

If[Head[lines] =!= List, lines = {lines, lines}];

If[ds === Automatic, nds = N[(s1 - s0)/(lines[[1]] - 1)]];

If[dt === Automatic, ndt = N[(t1 - t0)/(lines[[2]] - 1)]];

hg = Curves[e, {s, s0, s1, nds}, {t, t0, t1}, opts];

vg = Curves[e, {t, t0, t1, ndt}, {s, s0, s1}, opts];

Show[Graphics[Join[hg, vg]],

Evaluate[FilterRules[{opts}, Options[Graphics]]],

AspectRatio -> Automatic, Axes -> True]];

Curves[xy_, spread_, bounds_, opts___] :=

With[{curves = Table[{Re[xy], Im[xy]}, spread]},

ParametricPlot[curves, bounds, DisplayFunction -> Identity,

Evaluate[FilterRules[{opts}, Options[ParametricPlot]]]][[1]]];

指数函数

余弦函数

自定义的函数：倒数

Zeta 函数

复共轭的倒数

上面都是直角坐标下的图形，极坐标下的用法是完全相同的，不再举例说明。