Need sources to search for gamma function infinite series identities.

In summary, there are various resources available for finding infinite series summations for the gamma function, such as the Digital Library of Mathematical Functions and the Abramowitz and Stegun handbook. Gradshteyn and Ryzhik's 'Tables of Integrals, Series, and Products' is also a helpful reference. Searching through scientific journals may be challenging, so it is suggested to look at large lists of known series. The speed of convergence is important, but other factors may also be considered. The book by Whittaker and Watson is a valuable resource for finding various expansion formulas and references.
  • #1
mesa
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I have a hard time believing we only have the limited number of series I have seen so far especially considering how much broader mathematics is than I had thought just a short while ago.

Where should I search to find more infinite series summations for the gamma function? For example which journals would be good to check?
 
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  • #2
The Digital Library of Mathematical Functions is a website maintained by the National Institute of Standards and Technology. It is the modern successor to the old standby Abramowitz and Stegun book from 1964.

http://dlmf.nist.gov/

There is a chapter on the Gamma function.

Gradshteyn and Ryzhik, 'Tables of Integrals, Series, and Products' is also a good reference to have. As a handbook, it contains much information about special mathematical functions and series representations.
If you google carefully, copies of this work can be found online.
 
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  • #3
SteamKing said:
The Digital Library of Mathematical Functions is a website maintained by the National Institute of Standards and Technology. It is the modern successor to the old standby Abramowitz and Stegun book from 1964.

http://dlmf.nist.gov/

There is a chapter on the Gamma function.

Gradshteyn and Ryzhik, 'Tables of Integrals, Series, and Products' is also a good reference to have. As a handbook, it contains much information about special mathematical functions and series representations.
If you google carefully, copies of this work can be found online.

I should have mentioned I was told by another member to check out http://dlmf.nist.gov/
and the Abramowitz and Stegun handbook (which by the way is awesome! I need to get a copy for my bookshelf...) Aside from this all I have looked at is wolfram and wikipedia combined with some long hours searching Google.

Gradshteyn and Ryzhik, 'Tables of Integrals, Series, and Products' is a new suggestion. I checked the campus libraries and we seem to have several copies including some of the newer additions. I'll swing through and grab a copy between classes. Thanks for the suggestions!
 
  • #4
The Mathematica functions site lists a wealth of information of this type, and it is very well organized. This page, for example, lists 43 series representations of the gamma function Gamma(z).
 
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  • #5
phyzguy said:
The Mathematica functions site lists a wealth of information of this type, and it is very well organized. This page, for example, lists 43 series representations of the gamma function Gamma(z).

I came across that when exploring wolfram, it is the most comprehensive list I have seen yet. Mine isn't on their so the search continues.

Shouldn't I be checking the Scientific journals as well? Honestly I do not know where to start...
 
  • #6
I guess you could try. I don't know that there will be an unlimited, or even more than a few, infinite series representations of most functions.

Unless you are trying to demonstrate some mathematical property of a function which can only be gleaned from an infinite series representation, it would seem that one series is just as good as another.
 
  • #7
SteamKing said:
I guess you could try. I don't know that there will be an unlimited, or even more than a few, infinite series representations of most functions.

Okay.

SteamKing said:
Unless you are trying to demonstrate some mathematical property of a function which can only be gleaned from an infinite series representation, it would seem that one series is just as good as another.

Wouldn't a simpler and faster converging series typically be 'better' than a more complex slowly converging one?
 
  • #8
mesa said:
Shouldn't I be checking the Scientific journals as well? Honestly I do not know where to start...

Checking journals for your particular series is going to be difficult. It is unlikely something like the series itself would be present in the title, so you would probably need to search specifically for articles about the gamma function, the problem then becoming there are a lot of these! Further if it does appear in an article, this new series representation is likely not the focal point, so it becomes yet more challenging to find. This is why checking large lists of known series is more likely to be fruitful than searching the literature.

mesa said:
Wouldn't a simpler and faster converging series typically be 'better' than a more complex slowly converging one?

There is certainly some truth to this. To SteamKing's point, however, the mathematical community as a whole has largely moved on from questions like these. So focus on developing new series and how quickly they converge is kind of a fringe area. This could help you narrow your journal search though by looking at where people usually publish in this field.
 
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  • #9
mesa said:
Wouldn't a simpler and faster converging series typically be 'better' than a more complex slowly converging one?

Well, it depends. It you want to evaluate a function, certainly, a faster converging series is more desirable.

Other series, like the Gregory-Leibniz series for calculating pi, are notoriously slow to converge even to a handful of digits, but they can be memorized rather easily.

http://en.wikipedia.org/wiki/Pi#Rapidly_convergent_series
 
  • #10
SteamKing said:
Well, it depends. It you want to evaluate a function, certainly, a faster converging series is more desirable.

Other series, like the Gregory-Leibniz series for calculating pi, are notoriously slow to converge even to a handful of digits, but they can be memorized rather easily.

http://en.wikipedia.org/wiki/Pi#Rapidly_convergent_series

Makes sense.

I have seen the Gregory-Leibniz series before. As far as the 'rapidly convergent series' for Pi... well,
those series are just amazing!
 
  • #11
Look for a copy of https://www.amazon.com/dp/0521588073/?tag=pfamazon01-20 by Whittaker and Watson, either in a library or online for sale. This classic text (last revised in 1927) has an entire chapter on the Gamma Function and is loaded with various expansion formulas, plus references to older papers and books.
 
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  • #12
Petek said:
Look for a copy of https://www.amazon.com/dp/0521588073/?tag=pfamazon01-20 by Whittaker and Watson, either in a library or online for sale. This classic text (last revised in 1927) has an entire chapter on the Gamma Function and is loaded with various expansion formulas, plus references to older papers and books.

Our campus library has several copies of that book as well. I will be sure to pick one up while I am there!
 
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  • #13
jgens said:
Checking journals for your particular series is going to be difficult. It is unlikely something like the series itself would be present in the title, so you would probably need to search specifically for articles about the gamma function, the problem then becoming there are a lot of these! Further if it does appear in an article, this new series representation is likely not the focal point, so it becomes yet more challenging to find. This is why checking large lists of known series is more likely to be fruitful than searching the literature.

Good to know.

jgens said:
There is certainly some truth to this. To SteamKing's point, however, the mathematical community as a whole has largely moved on from questions like these. So focus on developing new series and how quickly they converge is kind of a fringe area. This could help you narrow your journal search though by looking at where people usually publish in this field.

So I have noticed, apparently I was born in the wrong century :biggrin:
 
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  • #15
SteamKing said:
A copy of this work is available from the Internet Archive:

https://archive.org/details/courseofmodernan00whit

Look at that, although I managed to pick up a copy of 'A Course in Modern Mathematics' from the campus library.

Unfortunately they have to pull Gradshteyn and Ryzhik's, 'Tables of Integrals, Series, and Products' from storage (all the shelf copies are currently out), any chance you have a link for this book? I'll try some Google searches and see what I can dig up.
 
  • #16
I can tell you there is a link to G & R, but I can't publish the link here without violating PF policy.
 
  • #17
SteamKing said:
I can tell you there is a link to G & R, but I can't publish the link here without violating PF policy.

Very good. If I can't find it the library will have a copy ready in a few days.
 

FAQ: Need sources to search for gamma function infinite series identities.

What is the gamma function?

The gamma function is a mathematical function that generalizes the factorial function to non-integer values. It is denoted by the Greek letter gamma (Γ) and is defined as Γ(x) = ∫0 tx-1e-t dt.

2. Why is the gamma function important?

The gamma function has many applications in mathematics, physics, and engineering. It is used to solve problems involving areas, volumes, and lengths of curves. It also plays a crucial role in probability theory and statistics, as well as in the study of special functions.

3. How can I search for gamma function infinite series identities?

There are several sources you can use to search for gamma function infinite series identities. Some of the most commonly used sources include mathematical databases such as MathSciNet and Zentralblatt MATH, as well as online resources such as Wolfram MathWorld and the Digital Library of Mathematical Functions.

4. Where can I find examples of gamma function infinite series identities?

You can find examples of gamma function infinite series identities in various mathematical textbooks and journals. Additionally, many online resources, such as the Online Encyclopedia of Integer Sequences, also provide extensive lists of known identities and their corresponding references.

5. Can I derive my own gamma function infinite series identities?

Yes, you can derive your own gamma function infinite series identities by using various mathematical techniques such as integration, substitution, and series manipulation. However, it is important to verify your results and ensure that they are not already known identities.

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