Integration Formula Explained: Li & Lj

[hash054]

Integration formula??

I am a graduate student and during my research I have come across this integration formula shows in attached image file. Can anyone help me make sense of this equation because i couldn't find any help from the literature regarding this equation.
Li and Lj are shape functions in this equations whose values Li= ( xj - x ) / ( xj - xi ) and Lj= ( x - xi ) / ( xj - xi )

 

[Mark44]

I am a graduate student and during my research I have come across this integration formula shows in attached image file. Can anyone help me make sense of this equation because i couldn't find any help from the literature regarding this equation.
Li and Lj are shape functions in this equations whose values Li= ( xj - x ) / ( xj - xi ) and Lj= ( x - xi ) / ( xj - xi )

Here's your integral, slightly modified (using a and b as limits of integration rather than the single l (letter 'l') of your thumbnail.

$$ \int_a^b L_i^{\alpha}~L_j^{\beta}dl = \frac{\alpha ! \beta !}{(\alpha + \beta + 1)!}l$$

 

[hash054]

thanks for replying .. I know this is the integral.. I was asking about how can we transform it into this factorial form.. any help in that regard?

 

[Mark44]

I put that in so people wouldn't have to open your thumbnail in another window.

Can you tell us any more about these shape functions? I'm assuming that i and j are indexes and alpha and beta are exponents. What are x_i and x_j?

Some context as to where this formula came up might be helpful as well.

 

[SteamKing]

The book 'Applied Finite Element Analysis' by Larry Segerlind gives a derivation of this integration formula. See pp. 70-73 of the text:

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=13&cad=rja&ved=0CGQQFjAM&url=ftp%3A%2F%2Fwebsrl.fsb.hr%2FPredavanja_vjezbe_programi_rokovi%2FMetoda%2520konacnih%2520elemenata%2FFinite%2520Element%2520Method%2520(FEM)%2520Collection%2FApplied%2520Finite%2520Element%2520Analysis%2520-%2520Larry%2520J.%2520Segerlind.pdf&ei=gh6pUsTlLMabrAHH8oD4BQ&usg=AFQjCNFnbNNQj_-jqq8fAxIfVMJ-EiJTNQ&sig2=jDsp_KOBXvsJSapsncKdpQ&bvm=bv.57799294,d.aWM

 

[Chestermiller]

This is almost certainly going to involve the gamma function. Substitute y = L_j, and (1-y)=L_i. I assume that it is being integrated between y=0 and y=1. Get Abramowitz and Stegan, and look up gamma functions. The integrals in terms of y are likely to be in there.

 

[SteamKing]

As a matter of fact, it does. You can find several copies of Abramowitz and Stegun online with Google.

 

[hash054]

thanks guys for replying.. I am grateful.. yet I have not taken a course in which gamma functions were included so a little help in evaluating eq. 6.16,17 would be appreciated!

 

[Chestermiller]

thanks guys for replying.. I am grateful.. yet I have not taken a course in which gamma functions were included so a little help in evaluating eq. 6.16,17 would be appreciated!

No problem. You need to get yourself a math book that covers gamma functions. Probably Kreyzig would have it; check out the table of contents on amazon. Otherwise, google gamma functions.

 

[hash054]

Got it... the equation 6.16,17 are derived from beta function β (z,w).. which has a relation with gamma function and ultimately in terms of factorial.. The books you guys recommended worked for me! thanks for the help.. now i can continue! :)