# Fourier transform of sine function

While solving the Fourier transformation of a sine wave (say $$h(t)=A\sin (2 \pi f_0 t)$$) in time domain, we get two peaks in frequency domain in frequency space with a factor of $$(A/2)j$$ with algebraic sum of delta function for $$f+f_0$$ and $$f-f_0$$ frequency, where $$j$$ is the imaginary unit.

My question is,

1. The Fourier Transform of odd function is imaginary in frequency domain, so what is its physical significance if it is a imaginary space.

2. As delta function have unit area what does $$A/2$$ means which comes under calculation. Does it mean delta function for two peaks have area $$A/2$$ now?

3. Why do we get two peaks such that sine wave has only one frequency. And what does it show physically for a physicist?

• Note that we use MathJax to typeset mathematics; you can find a good tutorial here. Commented May 17, 2021 at 10:14

For a simple $$\sin$$ or $$\cos$$ function, I find it easiest to understand by just remembering their definitions: $$A\sin(\omega t) = \frac{A}{2i}\left(e^{i\omega t} - e^{-i\omega t} \right)$$ Now you can clearly see where the two coefficients come from. You have $$a_{\omega} = \frac{A}{2i} = -\frac{A}{2}i$$ and $$a_{-\omega} = -\frac{A}{2i} = \frac{A}{2}i$$ So you can see how you need two coefficients to make a $$sin$$ wave out of a linear combination of $$e^{in\omega t}$$s.
For the integral transformation you then necessarily need delta functions to pick out exactly those two points from the complex plane: \begin{align*} \sin(t) \propto \int \Big(\delta(\omega-1) + \delta(\omega+1) \Big)e^{i\omega t}\mathrm{d}\omega = e^{it} + e^{-it} \end{align*}
• I'm not sure why you mention the $\delta$-function, a periodic signal can be reconstructed with the fourier series. Are you asking about the integral transform? I'm not sure if there's a good physical interpretation of the $i$. Maybe you find it useful to think of summing over $A \cos(\omega n t + \varphi)$, so instead of summing over $\sin$ and $\cos$ you sum over cosines with phase shifts which is where the $i$ comes in. Commented May 18, 2021 at 8:24