Finding representations of antiderivatives without a closed form

In summary, the conversation discusses techniques for finding series representations of integrals with no closed form, specifically for real functions. The subject falls under analysis and special functions such as elliptic and hypergeometric are important to learn. Asymptotic series expansion is one method, but there may be other methods as well. The complexity of these special functions may require further study and exploration.
  • #1
Manchot
473
4
I was wondering if anyone knew of any good books (or textbooks, or websites) which discuss finding series representations of integrals which exist, but don't have a closed form. I'm interested in the subject at the moment, but I haven't had much luck online. Furthermore, what branch of calculus would this fall under? I'd imagine that it would be some kind of analysis, but seeing as I've never taken an analysis course, I have no idea what type. (I've only taken Calculus I, II, III and Diff Eq). Thanks for the help!

EDIT: I forgot to say that I'm really only interested in the integrals of real functions. I guess that that's important.
 
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  • #2
Obviously, any analytic function can be represented by a power series which can always be integrated term by term within an interval of convergence.

There are various special methods, for example the asymptotic series expansion for the error function.

Learn about special functions (I think the best way to do this is with mathematica), which varioous integrands cani be reduced to.
 
  • #3
Either if you do learn about special functions (the most important i would consider to be the elliptical ones and the hypergeometric),u still may encounter integrals whose values cannot be expressed by neither of the functions "Mathematica" knows...

Daniel.
 
  • #4
As for power series, I'm not terribly interested in those, either. However, asymptotic series are one of my interests. I just now looked at the derivation for the asymptotic expansion for the error function, and was kind of disappointed, because it just used the "infinite integration by parts" method which I already figured out how to do. Are there any other such methods?

Yeah, I have been learning some of those special functions recently, but I'm curious as to how I should go about learning the elliptic integrals and hypergeometric functions. I just MathWorld'ed them and was kind of stunned by their complexity. Where should I even begin?
 

1. What is a closed form representation of an antiderivative?

A closed form representation of an antiderivative is an expression that can be written using a finite number of standard mathematical operations, such as addition, subtraction, multiplication, division, exponentiation, and logarithms. It does not involve any infinite series or special functions, and can be evaluated at any point without the use of numerical methods.

2. Why is finding representations of antiderivatives without a closed form challenging?

Finding representations of antiderivatives without a closed form can be challenging because not all functions have closed form antiderivatives. In fact, most functions do not have closed form antiderivatives. This means that for many functions, there is no simple expression that can be written for their antiderivative, making it difficult to find a representation without using numerical methods or approximations.

3. What are some methods for finding representations of antiderivatives without a closed form?

Some methods for finding representations of antiderivatives without a closed form include using integration by parts, substitution, partial fractions, and various integration techniques such as trigonometric substitutions and integration of rational functions. However, these methods may not always be successful in finding a closed form representation.

4. Can technology be used to find representations of antiderivatives without a closed form?

Yes, technology such as computer algebra systems can be used to find representations of antiderivatives without a closed form. These systems use algorithms to perform symbolic integration and can often find closed form representations for a wide range of functions. However, there may still be cases where a closed form representation cannot be found.

5. Why is finding a closed form representation of an antiderivative important?

Finding a closed form representation of an antiderivative can be important for various reasons. It allows for a deeper understanding of the function and its behavior, and it can also make further calculations and analysis easier. In addition, closed form representations are often preferred over numerical approximations as they are exact and not subject to rounding errors or other numerical inaccuracies.

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