Spectrum of Functions - Series Expansion
I n the kitchen of Physics one important mathematical recipe is series expansion of a function. In this context the most important series in usage are Taylor and Fourier series. Here I don’t want to bother my readers with the laws of continuity or convergence of series. I just want to say a bed-time story about series expansion and want to explore simple mathematical truth of writing a function as series. When I first come to know about Taylor series, I often wonder, how it is possible to write a function as a series. What is the mystery behind it? Can anyone tell me? In Taylor series we decompose any function in spectrum of $x^n$ where $n=0$ $or$ $+$ $ve$ $integer$, but is this the only way to express any function in series? And secondly what ensure the decomposition of the function in this type of series? Let's come to the second question first by taking two facts into account; two basic laws: Fact I: “Any function, which is defined for both $+$ $ve$ and $-$ $...