#2470 taylor(elliptic_kc(m), m, 0, 1) fails

open
nobody
5
2012-11-18
2012-09-19
Anonymous
No

Maxima cannot compute the Taylor expansion of elliptic_kc(m) for m = 0. Expansions about other points is ok.

I think the problem is that diff(elliptic_kc(m),m) at m = 0 produces a form that is indeterminate at 0. Maxima is unable to evaluate the limit as m approaches 0.

Discussion

• Aleksas
2012-09-20

From maxima help:
Function: elliptic_kc (m)
The complete elliptic integral of the first kind, defined as
integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

Example 1 Compute taylor(elliptic_kc(m), m, 0, 1)

(%i1) taylor(integrate(1/sqrt(1-m*sin(x)^2),x,0,%pi/2),m,0,1);
(%o1)/T/ %pi/2+((at(integrate(sin(x)^2,x,0,%pi/2),m=0))*m)/2+...
(%i2) ev(%,integrate);
(%o2)/R/ (%pi*m+4*%pi)/8
(%i3) taylor(%,m,0,1);
(%o3)/T/ %pi/2+(%pi*m)/8+...

Example 2 Compute taylor(elliptic_kc(m), m, 0, 3)

(%i4) taylor(integrate(1/sqrt(1-m*sin(x)^2),x,0,%pi/2),m,0,3)\$
(%i5) ev(%,nouns)\$
(%i6) taylor(%,m,0,3);
(%o6)/T/ %pi/2+(%pi*m)/8+(9*%pi*m^2)/128+(25*%pi*m^3)/512+...

best

Aleksas D