在[tex=3.0x1.357]hCUpMH37yix3aqPLXiFgJQ==[/tex] 上，将下列函数按勒让德多项式展开为广义傅里叶级数. [tex=3.786x1.357]ejyZgRYnBSH3MhBlrTb1fQ==[/tex]

2022-06-29

在[tex=3.0x1.357]hCUpMH37yix3aqPLXiFgJQ==[/tex] 上，将下列函数按勒让德多项式展开为广义傅里叶级数. [tex=3.786x1.357]ejyZgRYnBSH3MhBlrTb1fQ==[/tex]

答案：

解 令 [tex=7.571x2.714]JIP+HroQ0R9bofXCHGPx4iOeMXegmGUjC8MiC66M4q0+D357UnoXO6LORfZFjmAx[/tex]由系数公式有[tex=10.571x2.857]jqJTqb2jjT8ali1veRnkciOqLZTsDUgqXrw/ivyuruBrzDOWCYUxpXGIc+MfLV/l3A3hRrYpCZSiaRJdAXm6ZQ==[/tex]因为 [tex=1.143x1.357]M7eFZhSCOUN37Yx3DlAzjQ==[/tex]是偶函数，故当[tex=2.214x1.357]seSvsHO37BFuSu1T7j3ScA==[/tex]为奇函数，即当 [tex=9.643x1.214]HpXSXAmLJzou0GiAseLHUomcHEhDzszEwlbP1jkgbaA=[/tex]时于是[tex=38.214x11.786]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[/tex]        (1)而[tex=16.571x2.857]uxDza9E1eN/oNoo9UNLw6KXlTfA4ZRTO8p+75lVUZ0+B6wsb5ss4+79gLy1FObHx4iIlWHH9c39+4IKWObzWZ7eG93QP9qrVLvNKO/d9uNhqIqSdXbOS5LdTqt3l62MUItHuOHWw0W6Pl0Co8c2Rcg==[/tex]   (2)将(2)中的 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 换为 [tex=1.929x1.143]qMmLG3OT6I+UYFeehawKuA==[/tex]得到[tex=16.071x2.857]uxDza9E1eN/oNoo9UNLw6IAT3hhM8aALECA+T3zu1qDJAksZHIKAPIBZ6zorUT+Unf4wWejYyyY/EstZ0IXDo6mRwaiPDWDb41U5mgQDaTa2jl5bI8alk+9TBW9lf6nQ[/tex]                   (3)将(2)(3)式一并代入(1)式，得[tex=32.643x2.786]iZlgVI3kge8dm1VCSxGSNyoJMwdudJSufFbCbMaLoFjEFXEM/oFKgsPKeVAlFO5Tij6ke4j9dI0nmJ6xiLTSfu5E5jvV56p0c1bqZhxUS7Z9rb5cvBf/SdMv/sh5u4hZImu7IGO1qK0a7y025KSkxVTtIMSNoJpf6uFxwRPgKY73FycolpibnPkBogkpIk7HXhiMReebMeJmvabhyVVCuS8llcFRa6TgU4IYFmJZxoYbX0uHnVpDP1GU4i3B9IctntodVXrvm0k26t9+QQLTpA==[/tex]于是[tex=24.714x3.571]fOE7nlefpRhEzE34LqS1kIgWdyYXI6/xeoQX9G8eCyKti6w0aJILxDzm2Gtswyr035RjzOyg45kVx8tz/J1MUA+eCtH74+OCFH4jBHO0bN6pldGvieATJphQkw9/bUfNn0KXHGqN/gpgt7uIfpXD4gShPgL4HIzjLRsXc1X7lME=[/tex]

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举一反三

内容

0
在区间[tex=3.143x1.286]hkfcjGLYAjsvq0PDNlY+WPInsrrCkdBS5fglUa3LJfc=[/tex]内将函数[tex=12.143x4.071]3SXgFQTQxdgUfEIUxSXyK8++zDcRBay5a48Q2A03GLEhUhF00dYwa/2ozP+2QLLEV2DhvE2fWVxdq6x00qA37rmzicFxZRqGvbbjPY0iO8+ytEBsJbCbM/VWLlL95W4a[/tex]展开为傅里叶级数。
1
将下列以[tex=1.071x1.0]tieuzjBYrMcmxP3HXZSPGQ==[/tex]为周期的函数[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]展开为傅里叶级数，如果[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]在[tex=2.929x1.357]QpSc4Vs3d1MTNQAH70ziEw==[/tex]上的表达式：[tex=11.286x1.5]uoK9dXTY5b+zhHj119y5pCgzedituUCxRZojcLgLEJHLEJv3ATnVkUij7MXL+UY/[/tex].
2
将下列以[tex=1.071x1.0]tieuzjBYrMcmxP3HXZSPGQ==[/tex]为周期的函数[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]展开为傅里叶级数，如果[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]在[tex=2.929x1.357]QpSc4Vs3d1MTNQAH70ziEw==[/tex]上的表达式：[tex=11.071x1.5]IJwuJNbSgcLpUSCQjZhLKBJRwnnW1lXjwfuv04S+mWv3dyXfEVmq9L4aeKPnzkK6[/tex].
3
[tex=5.929x1.786]KzRyvoOd5QUNPEnu0Ofhq7pkC7Y+XLgeoo45Btrcc1A=[/tex]，在[tex=2.714x1.286]Z+IbHDMObsSvDLqoG2gghw==[/tex]上按勒让德多项式展开求三次最佳平方逼近多项式。
4
将下列以[tex=1.071x1.0]tieuzjBYrMcmxP3HXZSPGQ==[/tex]为周期的函数[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]展开为傅里叶级数，如果[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]在[tex=2.929x1.357]QpSc4Vs3d1MTNQAH70ziEw==[/tex]上的表达式：[tex=9.714x1.357]j0ikBUEGw4d2AEflw2o0Ie2DTZ7v5Ty0vqhh7iBref2PI92JfJwAF/7b4kOXYelP[/tex].