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Even or Odd Functions?

  1. Jan 24, 2014 #1
    Hey PF. I was wondering if anyone can help me figure out i can tell if certain functions are even or odd. For example, the function i*cos(ax)*sin(bx) when integrated with respect to x between -1/2 and 1/2 is equal to zero. I believe this means that it is even because it is symmetric around the y axis. To do the integration I used wolfram alpha, as this integral is beyond my current understanding, but I'm suppose to be able to do this problem without a computer apparently. This leads me to believe I should be able to tell that the function is even just by looking at it. Can somebody help me out here?
     
  2. jcsd
  3. Jan 24, 2014 #2

    tiny-tim

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    Hi PsychonautQQ! :smile:
    Yup! :rolleyes:

    Hint: what is cos(-ax) ? what is sin(-bx) ? :wink:
     
  4. Jan 24, 2014 #3

    PeroK

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    sine is odd and cosine is even, so the product is odd. So, the integral over [-1/2, 1/2] must be zero.

    Note that an integral being 0 doesn't imply the function is odd, as there are other ways for the integral to be 0.
     
  5. Jan 24, 2014 #4
    so you have an odd * even so the product is odd.. okay i'll take your word for it. The 'i' doesn't complicate the matter?
     
  6. Jan 24, 2014 #5

    tiny-tim

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    don't be silly!! :redface:

    never take anybody's word for any maths, check it yourself by working it out, or you'll never understand it or remember it

    TO PROVE: if f is even and g is odd, then f*g is odd

    PROOF … ? :smile:
     
  7. Jan 24, 2014 #6
    That said, if [itex]f:[-K,K]\to\mathbb R[/itex] is continuous, then [itex]f[/itex] is odd iff [itex]\int_{-x}^x f = 0[/itex] for every [itex]x\in [0,K][/itex].

    That'd be a cute problem set question in 1st-year calc, no?
     
  8. Jan 24, 2014 #7

    PeroK

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    Yes, very cute!
     
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