This series was inspired by a question that my wife asked me: calculate
nvenient value of that I want without changing the value of
. As shown in previous posts, substituting
yields the following simplification:
where I’ve assumed , the contour
in the complex plane is shown below (graphic courtesy of Mathworld),
and the positive constants and
are given by
,
.
Now we have the small matter of simplifying our expression for . Actually, this isn’t a small matter because Mathematica 10.1 is not able to simplify this expression much at all:
Fortunately, humans can still do some things that computers can’t. The numbers and
are chosen so that
and
are the roots of the denominator
. In other words,
,
These relationships will be very handy for simplifying our expression for :
To complete the calculation, we recall that
,
,
so that
,
,
and hence
since and
are both positive. Also,
,
so that
.
Finally,
.
Therefore,
.
1 Comment