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三角函数化简
In[14]:= TrigFactor[1/2 Cos[1] Cos[2 x] + 1/2 Sin[1] Sin[2 x]]
Out[14]= 1/2 Cos[1 - 2 x]
In[15]:= TrigExpand[1/2 Cos[1 - 2 x]]
Out[15]= 1/2 Cos[1] Cos[x]^2 + Cos[x] Sin[1] Sin[x] -
1/2 Cos[1] Sin[x]^2
In[23]:= DSolve[y''[x] - y'[x] - 6 y[x] == 0, y[x], x]
Out[23]= {{y[x] -> E^(-2 x) C[1] + E^(3 x) C[2]}}
画复平面上的单位圆周:
ParametricPlot[{Re[Exp[I z]], Im[Exp[I z]]}, {z, 0, 2 Pi}]
注意Real 与Re并不是一回事,更紧凑的写法:
ParametricPlot[Through[{Re, Im}[Exp[I [Theta]]]], {r, 0, 1}, {[Theta], 0, 2 Pi}]
相当于极坐标:
ParametricPlot[Through[{Re, Im}[r Exp[I [Theta]]]], {r, 0, 1}, {[Theta], 0, 2 Pi}, PlotStyle->None]
复指数函数映射:
ParametricPlot[ Through[{Re, Im}[ Exp[x + I y]]], {x, -1, 1}, {y, -1, 1 }, PlotStyle -> None]
华罗庚《从单位圆说起》中的第一个映射:
a = 0.01 + 0.3 I;
ParametricPlot[ Through[{Re, Im}[(r Exp[I [Theta]] - a)/(1 - Conjugate[a] r Exp[I [Theta]])]], {r, 0, 1}, {[Theta], 0, 2 Pi}, PlotStyle -> None, PlotRange -> All]
(华罗庚《从单位圆说起》中的第二个映射)
[Theta] = 3;
ParametricPlot[
Through[{Re, Im}[(Exp[I [Theta]] (x + I y))]], {x, -1, 1}, {y, -1,
1}, PlotStyle -> None]
单位圆周上的映射:
a = 1;
ParametricPlot[
Through[{Re,
Im}[(1 - a Exp[-I [Theta]])/(1 -
Conjugate[a] Exp[I [Theta]] )]], {r, 0, 1}, {[Theta], 0, 2},
PlotStyle -> None]
求复数的共轭:
Refine[Conjugate[r + b I], (r | b) [Element] Reals]
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