# I Defining Functions as Sums of Series

1. Apr 5, 2016

### Staff: Mentor

My Calculus 2 teacher's lecture slides say:

Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series.

I was just wondering how this was different from the basic functions that we've already worked with. Are they not defined as sums of series as well? If not, can they be?

2. Apr 5, 2016

### axmls

You can write many (most?) functions as an infinite sum, i.e. a Taylor series. I'm not sure entirely what the distinction is, but Bessel functions, for instance, cannot be written in terms of only elementary functions. They can be written as infinite sums or as integrals of elementary functions, but not with a straightforward combination of elementary functions. I suppose this is what differentiates a function that is "defined" as an infinite sum versus one that can simply be written as an infinite sum.

To me, though, I imagine the purpose of the notes was to express that sum functions can only be written as a sum, as opposed to dealing with the fact that most functions can be written as a sum.

3. Apr 5, 2016

### Samy_A

I also think like @axmls . What they probably try to convey is that functions like the Bessel functions are not elementary functions.

4. Apr 5, 2016

### Staff: Mentor

Just to clarify, a function as mentioned in the quote is a sum with an infinite number of terms; i.e., an infinite series.

As written, the sentence might be interpreted to say that such a function could be a sum of series, while a series is already an infinite sum.

5. Apr 5, 2016

### Staff: Mentor

Alright. Thanks for the comments all!