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奇延拓与偶延拓

2011-08-14 15:38:08| 分类：数学专题讲解 | 标签： |举报 |字号大中小订阅

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设函数f(x)定义在区间[0,a]上，则我们可以将f(x)延拓成为区间[-a,a]上的奇函数F(x)或偶函数G(x)。公式如下：

奇延拓的函数：F(x)=f(x) (当0<=x<=a)，F(x)= -f(-x) (当-a<=x<0)；

偶延拓的函数：G(x)=f(x) (当0<=x<=a)，G(x)= f(-x) (当-a<=x<0)。

例1 将函数f(x)=cos(x) (当0<=x<Pi/2)，f(x)=0 (当Pi/2<=x<=Pi) 分别延拓成区间[-Pi, Pi]上的奇函数和偶函数。

解

奇延拓：F(x)=f(x) (当0<=x<=Pi)，F(x)= -f(-x) (当-Pi<=x<0)；

偶延拓：G(x)=f(x) (当0<=x<=Pi)，G(x)= f(-x) (当-Pi<=x<0)。

图形如下：

f(x)的图形：

奇延拓与偶延拓 - calculus - 高等数学

奇延拓的图形：

奇延拓与偶延拓 - calculus - 高等数学

偶延拓的图形：

奇延拓与偶延拓 - calculus - 高等数学

作图的Mathematica程序：

f[x_] := Piecewise[{{Cos[x], 0 <= x < Pi/2}, {0, Pi/2 <= x <= Pi}}]
F[x_] := Piecewise[{{f[x], 0 <= x <= Pi}, {-f[-x], -Pi <= x < 0}}]
A = Plot[f[x], {x, 0, Pi}, PlotStyle -> {Red, AbsoluteThickness[3]}, Ticks -> {Range[-Pi, Pi, Pi/2], Range[-1, 1, 1/2]}]
B = Plot[F[x], {x, -Pi, Pi}, PlotStyle -> {Blue, AbsoluteThickness[2]}];
Show[B, A, Ticks -> {Range[-Pi, Pi, Pi/2], Range[-1, 1, 1/2]}]

f[x_] := Piecewise[{{Cos[x], 0 <= x < Pi/2}, {0, Pi/2 <= x <= Pi}}]

G[x_] := Piecewise[{{f[x], 0 <= x <= Pi}, {f[-x], -Pi <= x < 0}}]
A = Plot[f[x], {x, 0, Pi}, PlotStyle -> {Red, AbsoluteThickness[3]}, Ticks -> {Range[-Pi, Pi, Pi/2], Range[-1, 1, 1/2]}]
B = Plot[G[x], {x, -Pi, Pi}, PlotStyle -> {Blue, AbsoluteThickness[2]}];
Show[B, A, Ticks -> {Range[-Pi, Pi, Pi/2], Range[-1, 1, 1/2]}]

例2 将函数f(x)=cos(10x^3) (当0<=x<0.8)，f(x)=x^2 (当0.8<=x<=1.4) ，f(x)=sin(x) (当1.4<x<=2) 分别延拓成区间[-Pi, Pi]上的奇函数和偶函数。

解

奇延拓：F(x)=f(x) (当0<=x<=2)，F(x)= -f(-x) (当-2<=x<0)；

偶延拓：G(x)=f(x) (当0<=x<=2，G(x)= f(-x) (当-2=x<0)。

图形如下：

f(x)的图形：

奇延拓与偶延拓 - calculus - 高等数学

奇延拓的图形：

奇延拓与偶延拓 - calculus - 高等数学

偶延拓的图形：

奇延拓与偶延拓 - calculus - 高等数学

作图的Mathematica程序：

f[x_] := Piecewise[{{Cos[10*x^3], 0 <= x < 0.8}, {x^2, 0.8 <= x <= 1.4}, {Sin[x], 1.4 < x <= 2}}]
F[x_] := Piecewise[{{f[x], 0 <= x <= 2}, {-f[-x], -2 <= x < 0}}]
A = Plot[f[x], {x, 0, 2}, PlotRange -> All, PlotStyle -> {Red, AbsoluteThickness[3]}, Ticks -> {Range[-2, 2, 0.2], Range[-3, 4, 1]}]
B = Plot[F[x], {x, -2, 2}, PlotRange -> All, PlotStyle -> {Blue, AbsoluteThickness[2]}];
Show[B, A, Ticks -> {Range[-2, 2, 0.4], Range[-3, 4, 1]}]

f[x_] := Piecewise[{{Cos[10*x^3], 0 <= x < 0.8}, {x^2, 0.8 <= x <= 1.4}, {Sin[x], 1.4 < x <= 2}}]
G[x_] := Piecewise[{{f[x], 0 <= x <= 2}, {f[-x], -2 <= x < 0}}]
A = Plot[f[x], {x, 0, 2}, PlotRange -> All, PlotStyle -> {Red, AbsoluteThickness[3]}, Ticks -> {Range[-2, 2, 0.2], Range[-3, 4, 1]}]
B = Plot[G[x], {x, -2, 2}, PlotRange -> All, PlotStyle -> {Blue, AbsoluteThickness[2]}];
Show[B, A, Ticks -> {Range[-2, 2, 0.4], Range[-3, 4, 1]}]

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