Is This Function Even or Odd? Help Me Figure It Out!

[PsychonautQQ]

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?

 

[tiny-tim]

Hi PsychonautQQ!

This leads me to believe I should be able to tell that the function is even just by looking at it.

Yup!

Hint: what is cos(-ax) ? what is sin(-bx) ? ​

 

[PeroK]

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.

 

[PsychonautQQ]

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?

 

[tiny-tim]

so you have an odd * even so the product is odd.. okay i'll take your word for it.

don't be silly!

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 … ? ​

 

[economicsnerd]

Note that an integral being 0 doesn't imply the function is odd, as there are other ways for the integral to be 0.

That said, if $f:[-K,K]\to\mathbb R$ is continuous, then $f$ is odd iff $\int_{-x}^x f = 0$ for every $x\in [0,K]$.

That'd be a cute problem set question in 1st-year calc, no?

 

[PeroK]

Yes, very cute!