An interesting zero?

Posted Oct. 16 2009 by Paul Conant     We choose arbitrarily A and B as positive reals, neither of which is proportional to the number e.  We have     Ax + Bx - Cx = 0  We rewrite this as     exlnA + exlnB - exlnC  This gives     1 + xlnA + (xlnA)2/2! + (xlnA)3/3! + ...     1 + xlnB + (xlnB)2/2! + (xlnB)3/3! + ...   - 1 - xlnC - (xlnC)2/2! - (xlnC)3/3! - ...  which equals     1 + x(lnAB/C) + x2[(lnA)2 + (lnB)2 - (lnC)2]/2! + ...  That is,

0 = 1 + å¥j=0 xj/j![(lnA)j + (lnB)j - (lnC)j]  So we see the sigma sum equals -1 = eip  So we see that -1 can be expressed by an infinite family of infinite series.  Further, we may write

x = -åxj+1/j![(lnA)j + (lnB)j - (lnC)j]

and so any x may be so expressed.

This also holds for z = x + iy = reiu  And of course, we also have     c = 2/x + [lnAB] + x/2![(lnA)2 + (lnB)2] + x2/3![(lnA)3 + (lnB)3] ...       = 2/x + åj=1¥  xj-1/j![(lnA)j + (lnB)j]