MATLAB: Elliptic Integrals And Discrepancies with Published Data

elliptic integralsnot a bugpossible bug

I have noticed a rather significant discrepancy between the values of the elliptic integrals returned by ellipke() and those published in a number of texts. I am not sure if this is me not understanding the inputs or if it is genuinely wrong. I have checked these values against two texts as well as my own calculation based on Simpson's rule.
If anyone could offer some advice or tell me what I am doing wrong, it would be VERY appreciated.
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Reference 1: Lewis,'Vortex Element Method for Fluid Dynamic Analysis of Engineering Systems' 1991 pg 519-520.
Reference 2: Liepman, 'Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. pg 610-611. <http://www.convertit.com/Go/GovCon/Reference/AMS55.ASP?Res=150&Page=610&Submit=Go> _________________________________________________________________
For alpha=45 deg, k=sin(alpha)=0.707107
Reference 1 Gives: K=1.854074677301372 and E=1.350643881047676
Reference 2 Gives: K=1.854075 and E=1.350644
[K,E]=ellipke(0.707107); Gives: K=2.085974029127680 and E=1.237422393581531
Integrating Via Simpson's Rule Gives: K=1.854075629303571 and E=1.350644357747139
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My Simpsons Rule Code:
function [ K,E ] = FunEllipticIntegrals( phi )
%EllipticalIntegral1stKind Calculates the value of the elliptic integral of
%the first kind using simpsons rule.
tol=10^-6;
k=sind(phi);
i=1;
for a=0:tol:pi/2
  b=a+tol;
  [FunK1] = FunFirstkind(k,a);
  [FunK2] = FunFirstkind(k,(a+b)/2);
  [FunK3] = FunFirstkind(k,b);
  FunK(i) = (b-a)/6*(FunK1+4*FunK2+FunK3);
  [FunE1] = FunSecondkind(k,a);
  [FunE2] = FunSecondkind(k,(a+b)/2);
  [FunE3] = FunSecondkind(k,b);
  FunE(i) = (b-a)/6*(FunE1+4*FunE2+FunE3);
  i=i+1;
end
K=sum(FunK);
E=sum(FunE);
end
function [ FunK ] = FunFirstkind( k,alpha )
%FunFirstkind Elliptic Integral of 1st Kind Fuction
FunK=1/sqrt(1-k^2*sin(alpha)^2);
end
function [ FunE ] = FunSecondkind( k,alpha )
%FunFirstkind Elliptic Integral of 2nd Kind Fuction
FunE=sqrt(1-k^2*sin(alpha)^2);
end

Best Answer

Not a bug. (But you do get a +1 vote from me since your question was very clear and you provided enough info!)
Look at the bottom line of the definitions in:
doc ellipke
Some definitions of K and E use the elliptical modulus k instead of the parameter m. They are related as k^2 = m = sin(alpha)^2
checking this:
[K,E]=ellipke(sin(pi/4)^2)
K =
  1.8541
E =
  1.3506
We see your expected results.
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