A Series question
- To: mathgroup at smc.vnet.net
- Subject: [mg84731] A Series question
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Fri, 11 Jan 2008 22:06:55 -0500 (EST)
If we ask for Series[EllipticK[k^2], {k, 1, 0}],
we get a result involving Floor and Arg. That's understandable.
But suppose that we know that our k values will always be slightly less
than 1 and so we attempt to get a simpler result by assuming that k < 1:
In[3]:= Series[EllipticK[k^2], {k, 1, 0}, Assumptions -> k < 1]
Out[3]= (-I Pi/2 + 3/2 Log[2] - 1/2 Log[-1 + k]) + O[k - 1]^1
Although simpler, it's wrong since it is non-real complex. [Note that,
for -1 < k < 1, EllipticK[k^2] is real.]
However, there is a simple work-around:
In[4]:= Series[EllipticK[k^2], {k, 1, 0}, Assumptions -> -1 < k < 1]
Out[4]= (I Pi/2 + 3/2 Log[2] - 1/2 Log[-1 + k]) + O[k - 1]^1
Note that, compared to Out[3], the sign on the first term changed, and so
Out[4] is correct.
In[5]:= FullSimplify[%, -1 < k < 1]
Out[5]= 1/2 Log[-8/(-1 + k)] + O[k - 1]^1
which is clearly real for k < 1.
Should I have expected to have to assume -1 < k < 1 in order to get a
correct answer? I would suppose not. It seems to me that assuming k < 1
should have been adequate. If my supposition is correct, then Out[3] shows
a bug.
David