[math-fun] Here I go again.
Both http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html (18) and https://en.wikipedia.org/wiki/Elliptic_integral#Differential_equation give K'[k] == E[k]/k/(1 - k^2) - K[k]/k . Both sources use the k (modulus) notation vs Mathematica's m (parameter) notation. But I can't find *any* interpretation that makes this work. I see no similar claim in DLMF nor Borwein*2. ?? --rwg
* Bill Gosper <billgosper@gmail.com> [Aug 08. 2015 11:07]:
Both http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html (18) and https://en.wikipedia.org/wiki/Elliptic_integral#Differential_equation give K'[k] == E[k]/k/(1 - k^2) - K[k]/k .
This should be corrected (just did that) to d/dk K[k] == E[k]/k/(1 - k^2) - K[k]/k . (...if my relation (31.2-26a) on p.604 is correct). Apparently I lifted the 4 formulas shown from p.75 of Alfred Cardew Dixon: The elementary properties of the elliptic functions, with examples, Macmillan, (1894). http://www.archive.org/details/117736039 Best regards, jj
Both sources use the k (modulus) notation vs Mathematica's m (parameter) notation. But I can't find *any* interpretation that makes this work. I see no similar claim in DLMF nor Borwein*2. ?? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 2015-08-08 02:22, Joerg Arndt wrote:
* Bill Gosper <billgosper@gmail.com> [Aug 08. 2015 11:07]:
Both http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html (18) and https://en.wikipedia.org/wiki/Elliptic_integral#Differential_equation give K'[k] == E[k]/k/(1 - k^2) - K[k]/k .
This should be corrected (just did that) to d/dk K[k] == E[k]/k/(1 - k^2) - K[k]/k . (...if my relation (31.2-26a) on p.604 is correct).
Apparently I lifted the 4 formulas shown from p.75 of Alfred Cardew Dixon: The elementary properties of the elliptic functions, with examples, Macmillan, (1894). http://www.archive.org/details/117736039
Best regards, jj
Both sources use the k (modulus) notation vs Mathematica's m (parameter) notation. But I can't find *any* interpretation that makes this work. I see no similar claim in DLMF nor Borwein*2. ?? --rwg
Oh, for bleeping out loud! They meant Kdot rather than Kprime! In[898]:= D[OK[k], k] == OE[k]/k/(1 - k^2) - OK[k]/k Out[898]= (EllipticE[k^2] - (1 - k^2) EllipticK[k^2])/(k (1 - k^2)) == EllipticE[k^2]/(k (1 - k^2)) - EllipticK[k^2]/k In[899]:= Simplify[%] Out[899]= True (OK,OE := Old K := EllipticK[k^2], Old E:= EllipticE[k^2]), Why didn't I think to try that?? THANK YOU! Bill Ackerman: Your summary of the notational snafu is lucid and correct. However, I had already tried every imaginable correct and incorrect assumption (except confusing dot with prime) before making this appeal. --rwg
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Bill Gosper -
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rwg