How do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ?
What functions can be represented as power series?
3 Answers
A function can be represented as a power series if and only if it is complex differentiable in an open set. This follows from the general form of Taylor's theorem for complex functions.
Being real differentiable--even infinitely many times--is not enough, as the function $e^{-1/x^2}$ on the real line (equal to 0 at 0) is $C^\infty$ yet does not equal its power series expansion since all its derivatives at zero vanish. The reason is that the complexified version of the function is not even continuous at the origin.
-
3What does it mean to be complex differentiable in an open set? – 2010-07-23
-
3A function on an open subset of $\mathbb{C}$ to $\mathbb{C}$ is complex differentiable if the limit $\lim_{h \to 0} \frac{ f(z+h) - f(z)}{h}$ exists for all $z$ (analogous to the usual definition). It actually implies that derivatives of all orders exist, though. – 2010-07-23
To your question regarding radius of convergence, Wikipedia gives a good answer.
This is a very general question, as one can create all sorts of power series for different functions. (e.g. Taylor series, Laurent series, Fourier series).
To give the obvious example of Taylor series: a power series representation of a function can be found if the function is infinitely differentiable in the neighbourhood of the given point.
With all power series, you will need to find the recursion relation (formula giving a successive term from the current term) and then use the ratio test to solve for the value of the input variable that gives a ratio of convergence of 1.
-
0Could you possibly give an example of finding the radius of convergence for some simple function? – 2010-07-23
-
1It is not true that infinite differentiability implies the existence of a power series representation. Also, power series coefficients need not satisfy recurrence relations. – 2010-12-10
-
0@Jonas: Let me clarify, by infinitely differentiable I mean that the n'th derivative does not blow up. – 2010-12-10
-
1@Noldorin: I guess, to be more precise, you mean that the remainder term in Taylor's theorem goes to zero in an interval. However, that is not what infinite differentiability usually means. Infinite differentiability usually means that all derivatives are everywhere defined, and that is what I meant in my comment. – 2010-12-10
-
0If any derivative goes to infinity, then surely the Taylor series is invalid? – 2010-12-10
-
0If the function is infinitely differentiable on say $(-a,a)$, then each of its derivatives will be continuous, and therefore bounded on each compact subinterval $[-b,b]$. But unless there is some control over the growth of the derivatives on $[-b,b]$, one may not be able to say that the Taylor series (which can always be *defined* for an infinitely differentiable function) actually converges to the function on $[-b,b]$. A standard example is $f(x)=e^{-1/x^2}$ when $x\neq0$, $f(0)=0$. $f^{(n)}(0)=0$ for all $n$, so its Maclaurin series is everywhere $0$, & the remainder is always $f(x)$. – 2010-12-10
-
1The converse to your second paragraph holds: Analytic functions, that is, functions that have power series expansions about each point in their domains, are the nicest examples of infinitely differentiable functions. The answer at http://math.stackexchange.com/questions/12989/ refers to an example that shows that an infinitely differentiable function need not equal its Taylor series on an interval about *any* point, and http://en.wikipedia.org/wiki/Taylor%27s_theorem#Estimates_of_the_remainder discusses criteria to determine that the remainder goes to zero, meaning the series converges to f. – 2010-12-10
-
0@Jonas: Wow, that's some very subtle maths. Thanks for the info... At least, not something I really need to worry about as a physicist. :) – 2010-12-10
-
1@Noldorin: You're welcome. It may be something you don't need to worry about. However, I wouldn't be surprised if you come across smooth nonanalytic functions in physics. E.g., smooth functions with compact support are never analytic, and they come up in many areas of mathematics with applications (smooth manifolds and differential equations for instance). See also http://en.wikipedia.org/wiki/Bump_function – 2010-12-10
-
0@Jonas: You're probably right; though I imagine if/when they do become relevant, it will be at graduate level. I've probably already encountered non-analytic functions in my studies; just never had to actual think about power series expansions in relation to them! – 2010-12-10