[复变函数与积分变换].焦红伟&尹景本.文字版.PDF.pdf

2 2 2 2 t t t t f x x f x t t f x t x t 越? t f x

Info icon This preview shows pages 132–134. Sign up to view the full content.

( ) , , 2 2 ( ) ( ) , , 2 2 T T T T f x x f x T T f x T x ∈ − = + ∈ − T 越大, ( ) T f x ( ) f x 相等的范围也越大,这表明当 T → + 时,周期函数 ( ) T f x 便可以转 化为 ( ) f x ,即有 lim ( ) ( ) T T f x f x →+ = 这样,在式 (7.2) 中令 T → + 时,结果就可以看成是 ( ) f x 的展开式,即 2 i i 2 1 ( ) lim ( )e d e n n T x T T T n f x f T ω τ ω τ τ + →+∞ =− = n 取一切整数时, n w 所对应的点便均匀地分布在整个数轴上 . 若取两个相邻点的距离 ω 表示,即 1 2 π n n T ω ω ω = = ,或 2 π T ω = 则当 T → + ∞时,有 0 ω ,所以上式又可以写为
Image of page 132

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

7 傅里叶变换 · 127 · · 127 · 2 i i 2 0 1 ( ) lim ( )e d e 2 π n n T x T T n f x f ω τ ω ω τ τ ω + =− = (7.3) x 固定时, 2 i i 2 1 ( )e d e 2 π T x T T f ωτ ω τ τ 是参数 ω 的函数,记为 ( ) T Φ ω ,即 2 i i 2 1 ( ) ( )e d e 2 π T x T T T Φ f ωτ ω ω τ τ = 利用 ( ) T Φ ω 可将式 (7.3) 写成 0 ( ) lim ( ) T n n f x ω Φ ω ω + =− = 很明显,当 0 ω 时,即 T → + 时,有 ( ) ( ) T Φ ω Φ ω 这里, ( ) i i 1 ( )e d e 2 π x f ωτ ω Φ ω τ τ + = 从而 ( ) f x 可以看作是 ( ) Φ ω ( ) , + 上的积分 ( ) ( )d f x Φ ω ω + = ,即 i i 1 ( ) ( )e d e d 2 π x f x f ωτ ω τ τ ω + + = (7.4) 这个公式称为函数 ( ) f x 的傅里叶积分公式(简称傅里叶积分公式).应该指出,上式只 是由式 (7.3) 式的右端从形式上推出来的,是不严格的 . 至于一个非周期函数 ( ) f x 在什么条 件下,可以用傅里叶积分公式表示,有下面的定理 . 傅里叶积分定理 f ( t ) ( -∞, + ) 上满足: (1) 在任一有限区间上满足狄利克雷条件; (2) 在无限区间 ( -∞,+∞ ) 上,绝对可积 ( | ( ) | d f t t + 收敛 ) ;则有 i j ( ) ( ) 1 ( )e d e d 1 2 π [ ( 0) ( 0) ( ) 2 x f x f f x f x ωτ ω τ τ ω + + = + + 在连续点上 在间断点上 (7.5) 这个式子称为傅里叶积分的复指数形式,利用欧拉公式,可将它转化为三角形式 . 因为 ( ) i i i 1 ( ) ( )e d e d 2 π 1 ( )e d d 2 π 1 1 ( )cos ( )d d i ( )sin ( )d d 2 π 2 π x x f x f f f x f x ωτ ω ω τ τ τ ω τ τ ω τ ω τ τ ω τ ω τ τ ω + + + + + + + + = = = + 考虑到积分 ( )sin ( )d f x τ ω τ τ + ω 的奇函数,就有 ( )sin ( )d d 0 f x τ ω τ τ ω + + =
Image of page 133
Image of page 134
This is the end of the preview. Sign up to access the rest of the document.
  • Winter '16
  • Gilles Lamothe

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern