# Even or Odd Functions?

1. Jan 24, 2014

### 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?

2. Jan 24, 2014

### tiny-tim

Hi PsychonautQQ!
Yup!

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

3. Jan 24, 2014

### 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.

4. Jan 24, 2014

### 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?

5. Jan 24, 2014

### tiny-tim

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 … ?

6. Jan 24, 2014

### economicsnerd

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?

7. Jan 24, 2014

### PeroK

Yes, very cute!