Spectrum of Functions - Series Expansion

In 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 $-$ $ve$ of the variable, can be expressible in terms of two functions, one odd and another even function.”

$f(x)=O(x)+E(x)$

$O(x)=\frac{[f(x)-f(-x)]}{2}$        and         $E(x)=\frac{[f(x)+f(-x)]}{2}$

Here $O(-x)=-O(x)$, an odd function and $E(-x)=E(x)$, an even function. This law builds the heart of series expansion.

If $f(x)$ is an even function its odd part will turn out to be ‘0’ and same for an odd function its even part will be ‘zero’. You can try the rules provided here for finding odd and even part and it will turn that the statement provided here is perfectly true.

Fact II: Linear superposition of functions. Take two such combinations $\sum_{n=0}^r a_n \Psi_n(x)$ and $\sum_{n=0}^r b_n \Phi_n(x)$ and a condition is imposed that all $\Psi_n$'s are odd and all $\Phi_n$’s are even function. So now assign the first polynomial as $O_n(x)$ and the second as $E_n(x)$.

$O_n(x)=\sum_{n=0}^r a_n \Psi_n(x)$

$E_n(x)=\sum_{n=0}^r b_n \Phi_n(x)$

So combining both odd and even superimposed function, we have $L_n(x)=\sum_{n=0}^r a_n \Psi_n(x)+b_n \Phi_n$; as a polynomial (as it contains finite terms) and its corresponding series is $f(x)=\sum_{n=0}^r a_n \Psi_n+b_n \Phi_n$ (as it contains infinite terms) and must be endowed with the law of convergence to express the function with this series (study of convergence is excluded in this article).

With this two simple law we just reach up to the core formulation of problem of series expansion, which is actually built with law of superposition. Now $a_n$ and $b_n$’s are unknown coefficients. I want to call them “Weightage Factors”, which indicates contribution of a specific $\Psi_n$ or $\Phi_n$ in spectrum of $f(x)$. $\Psi_n$ and $\Phi_n$ will be termed as “Basis Functions”, just bringing the analogy from basis of vector space. Hence the problem reduces to finding of weightage factors. Now there are two main stream way to find these weightage factors:

• Finding of weightage factor by differentiating as used in series solution and Taylor series. 
• Finding of weightage factor by integration (using the property of orthogonality of functions) as used in Fourier series.

1. Taylor & Maclaurin Series: Here our choice for basis function is $x^n$, and $\Psi_n=x^{(2p+1)}$, odd functions and $\Phi_n=x^{2p}$, even functions; where $p=0$, $1$, $2$, $3$, $...$

$f(x)=\sum_{p=0} a_p x^{2p}+b_p x^{(2p+1)}$$=\sum_{n=0} c_n x^n$

Differentiating, We get

$f^{\prime}(x)=\sum_{n=1} n c_n x^{(n-1)}$ and

$f^{\prime\prime(x)}=\sum_{n=2} n (n-1) c_n x^{(n-1)}$ and

$f^{\prime\prime\prime(x)}=\sum_{n=3} n (n-1) (n-2) c_n x^{(n-2)}$ and so on...

Hence $f^{\prime}(0)=c_1$; $f^{\prime\prime}(x)=(1*2) c_2=2 c_2$; $f^{\prime\prime\prime}(x)=(1*2*3) c_3=6 c_3$; $f^{\prime\prime\prime}(x)=(1*2*3*4) c_4 =24 c_4$ and so on this determines the $c_n$;s or the expansion weightage factors and given by

$c_n=\frac{f^{n}(0)}{n!}$

Here we need the condition that function must have the continuous deferential up to order '$n$' in the region of expansion. As we just can’t continue up to infinite, we have to stop somewhere and truncate the series and this generates a reminder term to reduce the error. Actually here we expand $f(x)$ about the point $x=0$, and this is a special case of Taylor expansion, known as Maclaurin series.

2. Fourier Series: Now if we wish to express a function in light of periodicity then we can chose the prototype periodic functions $sin(nx)$ [odd function] and $cos(nx)$ [even function] as basis functions. So from general series format we can write it as

$f(x)=\sum_{n=0} a_n sin(nx)+b_n cos(nx)$

Now one can try the previous method to find the coefficients but the process will not work at all. We have to use the orthogonality condition of $sin(nx)$ and $cos(nx)$ functions within the region $-\pi$ to $\pi$, is given by

$\int_{-\pi}^{\pi} sin(nx) sin(mx) dx=\{ \begin{array}{1,1} \pi \delta_{mn} \qquad \quad \text{for $n \neq 0$, $m \neq 0$}\\ 0 \qquad \qquad \text{for $n=0$, $m=0$}\end{array}$

$\int_{-\pi}^{\pi} sin(nx) cos(mx) dx=0 \qquad \qquad \quad \text{for all $n$ and $m$}$

$\int_{-\pi}^{\pi} cos(nx) cos(mx) dx=\{ \begin{array}{1,1} \pi \delta_{mn} \qquad \quad \text{for $n \neq 0$, $m \neq 0$}\\ 2\pi \qquad \qquad \text{for $n=0$, $m=0$}\end{array}$

So, we get 

$a_n=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) sin(nx)dx \qquad \text{and}$

$b_0=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x)dx \qquad \qquad \qquad \text{and}$

$b_n=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) cos(nx)dx \quad \quad n \neq 0$

Here another two important observations, if we express $sin(nx)$ and $cos(nx)$ as spectrum of $x^n$ , we can able to get back an expression similar to Taylor series.

And if a $f(x)$ is an even function then only $b_n$’s will survive, all $a_n$’s will die out to ‘0’ (as even functions get an ‘0’ odd counterpart and $a_n$’s indicate the weightage factors for odd functions) and again if f(x) is an odd function all $b_n$’s will die out to ‘0’ (and logic will be similar to the previous one).

Surely this is not the end; we can have other alternative except these daily used well known series:

3. Using orthogonal polynomial to get series expansion: Both of the above series are very much familiar to us. Here in the previous section we have already introduced orthogonality condition to determine the coefficients. Now use this same property to expand function through an orthogonal polynomial. Consider the orthogonality relation of Legendre Polynomial $P_n(x)$ [Here if ‘$n$’ is even or ‘0’ $P_n(x)$ is an even function, else $P_n(x)$ is odd one.]

$\int_{-1}^{1} P_n(x) P_m(x) dx=\frac{2}{(2n+1)} \delta_{mn}$

So if we write $f(x)=\sum_{n=0} a_n P_n(x)$ and then $a_n$'s will be given by

$a_n=\frac{(2n+1)}{2} \int_{-1}^{1} f(x) P_n(x) dx$

As

$\int_{-1}^{1} f(x) P_m(x) dx=\sum_{n=0}a_n \int_{-1}^{1} P_n(x) P_m(x) dx=\frac{2a_n}{(2n+1)}$

Now we get the answer to our first question that we can create the spectrum of a function taking two set of functions ${\Psi_n}$ and ${\Phi_n}$ and also keeping it in mind that using some properties or tricky method we can at least find all the weightage factors, i.e. if we modify Fourier series like

$f(x)=\sum_{n=0} a_n sin^n(x)+b_n cos^n(x)$

Then the first condition for write a function in series is satisfied but the integration become terrifically tough and not all the coefficients are determinable.

You also use numerical method to express a function in series. That is an elaborated topic to introduce in this short note but interesting piece to work out. Hope you will try it out.

So ‘splitting of a function in an odd and even function’, ‘liner superposition’, ‘series solution’ and ‘orthogonality relation’ is the “basis” that “span” the group of series expansion. Here my story come to an end, get back to sleep my dear.

References:

[1] Common Sense; Publisher: Logical Human Brain [Don’t try to find this References in library]
[2] Mathematical Method for Physicists; Author: Arfken and Weber; Publisher: Elsevier

© Author: Krishanu Das, This article was published in departmental magazine - "Horizon", Publisher: Department of Physics, St. Xavier's College, Kolkata (Autonomous) in 2011. Here this article is uploaded as a knowledge base on public interest.
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