Is a Closed-Form Solution Possible for the Integral of e^cos(x)?

[Si14]

Homework Statement

I want to solve this integral without using series expansion. The answer should be in a closed form. I wonder if this is possible?

Homework Equations

The Attempt at a Solution

I used numerical methods and was able to solve it numerically for a given interval. However, I need to solve it without using numerical methods and without using series expansion.

Thanks.

 

[DryRun]

Try using substitution: Let y = cos x

 

[Si14]

sharks, it seems that it gets more complicated!
I stuck immediately after substitution.

 

[Ray Vickson]

Homework Statement

I want to solve this integral without using series expansion. The answer should be in a closed form. I wonder if this is possible?

Homework Equations

The Attempt at a Solution

I used numerical methods and was able to solve it numerically for a given interval. However, I need to solve it without using numerical methods and without using series expansion.


Thanks.

I suspect it is not doable in terms of elementary functions. Neither Wolfram Alpha nor Maple 11 can find non-series expressions for the indefinite integral. You might try converting it so some known special (but non-elementary) function, perhaps by using integration by parts and/or substitution methods.

RGV

 

[Dickfore]

The primitive function is not elementary. Do you want the definite or the indefinite integral? Because, if you look at the integral representation of the Bessel function, you might find a similarity.

 

[DryRun]

sharks, it seems that it gets more complicated!
I stuck immediately after substitution.

\int e^y.\frac{-1}{\sqrt{1-y^2}}dyThen, use integration by parts.

 

[Si14]

I wonder if it is possible to solve the indefinite integral? I assume the answer to the definite one should be close to the answer I get with numerical methods.

 

[Dickfore]

As I said, the primitive function is not elementary
Wolfram Alpha

 

[Si14]

I checked wolframalpha. However, it gives a series expansion which I can not use.
I wonder if the integration by parts suggested by sharks is doable?

 

[Dickfore]

You obviouly don't know what primitive or elementary means. Did you see the first sentence mentioned in wolfram alpha? It tells an even more stringent condition, in terms of "standard mathematical functions", which includes some non-elementary functions (including Bessel functions).

 

[DryRun]

I checked wolframalpha. However, it gives a series expansion which I can not use.
I wonder if the integration by parts suggested by sharks is doable?

Try $u=e^y$ and $dv=\frac{-1}{\sqrt{1-y^2}}$
Then, $du=e^y$ and $v=\cos^{-1}y$

 

[Ray Vickson]

I checked wolframalpha. However, it gives a series expansion which I can not use.
I wonder if the integration by parts suggested by sharks is doable?

It won't get you anywhere; the integral is non-elementary, and no amount of manipulation will change that fact. However, you might try to re-express the indefinite integral in terms of some already-defined special functions (as already suggested).

RGV