Series representation for this integral

In summary, it seems like the person is trying to find a series representation for a given mathematical expression and is wondering if there is a specific name for this function. They have found a series representation for a similar expression in a reference book, but it does not fit their given expression. They are seeking help in finding a solution and suggest trying to write power series representations and integrating term by term.
  • #1
ConfusedCat
3
0
I am trying to find a series representation for the following expression
$$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant.

Is there a name for this function?

I found a series representation for $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ in Table of Integrals, Series and Products by Gradshteyn and Ryzhik, involving parabolic cylinder function, but nothing that fits the first expression.

Any help would be appreciated.
 
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  • #2
I would try writing power series representations for the exponential and [tex](b+ x)^{-n}[/tex], multiply the series, then integrate term by term.
 

FAQ: Series representation for this integral

What is a series representation for an integral?

A series representation for an integral is a way to express an integral as an infinite sum of simpler functions, typically polynomials. It allows us to approximate the value of the integral using a finite number of terms from the series.

How is a series representation for an integral derived?

A series representation for an integral is derived using a technique called power series expansion, which involves breaking down the original function into simpler components and expressing them as a sum of powers of x. This series can then be integrated term by term to obtain a series representation for the integral.

What are the advantages of using a series representation for an integral?

One advantage of using a series representation for an integral is that it allows us to approximate the value of the integral with greater accuracy by using more terms from the series. Additionally, it can also help us to evaluate integrals that are difficult or impossible to evaluate using traditional methods.

Can a series representation for an integral be used to find exact values?

No, a series representation for an integral can only provide an approximation of the integral's value. In order to find the exact value, we would need to use other methods such as the Fundamental Theorem of Calculus or numerical integration techniques.

Are there any limitations to using a series representation for an integral?

Yes, there are limitations to using a series representation for an integral. It may not converge for all values of x, and the series may not accurately represent the original function for all values of x. Additionally, the process of deriving a series representation can be time-consuming and may not always be feasible for more complex integrals.

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