Power series, Taylor series and Functions
Since recently I am dealing with numerical solution of differential equation, I recalled that not long time ago I’ve actually learned about and Euler’s equation. They should be close related to the differential equation topic. So let’s review some of the conclusions I learned from the book: “オイラーの公式がわかる”.
(A series doesn’t need to have a variable in it: a infinite series of numbers can be a series.)
In mathematics, a power series (in one variable) is an infinite series of the form:
where represents the coefficient of the nth term and c is a constant. In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form:
Use a power series to generate a function
Suppose we have an arbitrary function . We can use a power series to approximate the function in the following way:
The power series takes the form:
Since should equal to
In a similar fashion, we differentiate the function once and should equal to
Repeat this algorithm one more time and we will find:
Finally the general form gives:
In fact, we just derived the Taylor series near 0 for .
Use such a technique, we can find the Taylor series of and (again, near 0):
(The fact that is odd and is even could also be found out from right hand side!)