Book recommendations for fourier series.

Question

I am a physics Student, i have finished both volumes of TOM M.APOSTOL calculus, but it doesn't have any chapter related to Fourier series, can you suggest me a book through which i can directly refer to Fourier series chapter with some what similar style like apostol.

Answer 1

The book Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima explains Fourier series. I recommend checking it out. You might also take a look at Fourier Series and Boundary Value Problems by Brown and Churchill. Mathematical Methods in the Physical Sciences by Boas is also worth a look.

The function $e^{2\pi i t}$ makes one revolution as $t$ goes from $0$ to $1$. It is the very essence of periodic motion. The function$e^{2 \pi i k t}$ completes $k$ revolutions as $t$ goes from $0$ to $1$. It is the essence of periodic motion at a higher frequency. Given a complex-valued function $f$ with period $1$ (perhaps $f(t)$ is the position of a planet at time $t$), a deep idea is to try to represent $f$ as a combination of pure frequencies: $$ \tag{1} f(t) = \sum_{k \in \mathbb Z} c_k e^{2 \pi i k t}. $$ This is the idea behind Fourier series. The numbers $c_k$ are called the Fourier coefficients of the function $f$. To compute $c_j$, use Fourier's trick: multiply both sides of (1) by $e^{-2\pi i j t}$, integrate from $0$ to $1$, and observe the wonderful cancellation that occurs.

Amazingly, any reasonably nice complex-valued function $f$ with period $1$ can be represented as in (1). Ancient astronomers benefited from this fact, unknowingly, when they used epicycles to describe the motions of planets.

Answer 2

You would have a rigorous introduction from Apostol. So I would recommend a solid book in Fourier Analysis. One that comes to mind is

Fourier Analysis: An Introduction (Princeton Lectures in Analysis, Volume 1) by Elias M. Stein, Rami Shakarchi.