Representation of Functions as Power Series

Transcription

1 Representation of Functions as Power Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Functions as Power Series Today /

2 Introduction In this section and the next, we develop several techniques to help us represent a function as a power series. More precisely, given a function f (x), we will try to find a power series c n (x a) n such that f (x) = c n (x a) n. The techniques involved are substitution, differentiation and integration. We will apply these techniques to known series, to derive new series representations. So far, this is the only series representation we know: x = + x + x 2 + x in (, ) = x n Philippe B. Laval (KSU) Functions as Power Series Today 2 /

3 Substitution We derive the series for a given function using another function for which we already have a power series representation. Then, we do the following: Figure out which substitution can be applied to transform the function for which we know the series representation to the function for which we want a series representation. 2 Apply the same substitution to the known series. This will give us the series representation we wanted. 3 The domain of the new function is obtained by applying the same substitution to the domain of the known series. Philippe B. Laval (KSU) Functions as Power Series Today 3 /

4 Substitution Find a power series representation for f (x) = + x Find a power series representation for 2 + x and find its domain. and find its domain. Use the previous example to find a power series representation for Find a power series representation for e x 2 if e x = x n n! x x. in (, ). Philippe B. Laval (KSU) Functions as Power Series Today 4 /

5 Differentiation and Integration of Power Series Theorem If the power series c n (x a) n has a radius of convergence R > 0 then the function defined by f (x) = c n (x a) n is differentiable (hence) continuous on (a R, a + R) and f (x) = nc n (x a) n (series can be differentiated term by 2 n= term). c n (x a) n+ f (x) dx = C + (series can be integrated term by n + term). 3 In both cases the radius of convergence is preserved. However, convergence at the endpoints must be investigated every time. Philippe B. Laval (KSU) Functions as Power Series Today 5 /

6 Differentiation and Integration of Power Series This theorem simply says that the sum rule for derivatives and integrals also applies to power series. Remember that a power series is a sum, but it is an infinite sum. So, in general, the results we know for finite sums do not apply to infinite sums. The theorem above says that it does in the case of differentiation and integration of infinite series. Given that a power series representation for f (x) is f (x) = + x + x 2 + x = x n find a power series representation for f (x) and f (x) dx. Philippe B. Laval (KSU) Functions as Power Series Today 6 /

7 Finding Series Representation by Differentiation This time, we find the series representation of a given series by differentiating the power series of a known function. More precisely, if f (x) = g (x), and if we have a series representation for f and need one for g, we simply differentiate the series representation of f. The theorem above tells us that the radius of convergence will be the same. However, we will have to check the endpoints. Find a series representation for, find the interval of convergence. 2 ( x) Suppose you know that a series representation for sin x is sin x = x x 3 3! + x 5 5! x 7 7! +... = ( ) n x 2n+. Find a power series (2n + )! representation for cos x. Philippe B. Laval (KSU) Functions as Power Series Today 7 /

8 Finding Series Representation by Integration This time we find the power series representation of a function by integrating the power series representation of a known function. If g (x) = f (x) dx and we know a power series representation for f (x), we can get a series representation for g (x) by integrating the series representation of f. Find a power series representation for ln ( x). Find a power series representation for tan x. Philippe B. Laval (KSU) Functions as Power Series Today 8 /

9 Things to Know Be able to find the series representation of a function by substitution, integration or differentiation. Remember the power series representations we found in this section and when the representation is valid. Philippe B. Laval (KSU) Functions as Power Series Today 9 /

10 Series representations Known so Far x = x n in (, ) + x = ( ) n x n in (, ) + x 2 = ( ) n x 2n in (, ) ( x) 2 = ln ( x) = tan x = nx n in (, ) n= x n+ n + ( ) n x 2n+ 2n + in [, ) in [, ] Philippe B. Laval (KSU) Functions as Power Series Today 0 /

11 Exercises See the problems at the end of my notes on representation of functions as power series. Philippe B. Laval (KSU) Functions as Power Series Today /