Find a fourier transform of f ( x ) = cos ⁡ ( x ) x This is what I have: F ( ω ) = ∫ − ∞ ∞ 1 2 x ( e i x + e − i x ) e − i ω x d x F ( ω ) = 1 2 ∫ − ∞ ∞ e i ( 1 − ω ) x x d x + 1 2 ∫ − ∞ ∞ e i ( − 1 − ω ) x x d xUsing integration in complex plane almost in the same manner as has been used in this example (I use fourier transform with minus sign), I obtain: I 1 = i π for ω &lt; 1 and − i π for ω &gt; 1 I 2 = i π for ω &lt; − 1 and − i π for ω &gt; − 1where I 1 and I 2 are first and second integral respectively

Question

Find a fourier transform of   f  (  x  )  =            cos      ⁡      (      x      )        x  This is what I have:  F  (  ω  )  =      ∫          −      ∞        ∞        1          2      x        (      e          i      x        +      e          −      i      x        )      e          −      i      ω      x        d  x  F  (  ω  )  =      1    2        ∫          −      ∞              ∞                  e              i        (        1        −        ω        )        x              x    d  x  +      1    2        ∫          −      ∞              ∞                  e              i        (        −        1        −        ω        )        x              x    d  xUsing integration in complex plane almost in the same manner as has been used in this example (I use fourier transform with minus sign), I obtain:      I    1    =  i  π for   ω  &amp;lt;  1 and   −  i  π for   ω  &amp;gt;  1      I    2    =  i  π for   ω  &amp;lt;  −  1 and   −  i  π for   ω  &amp;gt;  −  1where       I    1   and       I    2   are first and second integral respectively

Eric Solis · Accepted Answer

You are right! but wolfram uses as Fourier transform the following definition:  F  (  ω  )  =      1          2      π            ∫          −      ∞        ∞    f  (  t  )            e              i      ω      t            d    t

Need to find a fourier transform of f(x)=cos(x)/x.

Answered question

Answer & Explanation

New Questions in Calculus 1