> From: Joshua Zucker <zucker@csli.Stanford.EDU> > To: Multiple recipients of list <email@example.com> > Subject: Calculator > Date: Thursday, July 31, 1997 4:45 PM > > log(-1) gives an answer? How does it do that? (2n+1)*i*pi ? > > Or does it just give i*pi as the only answer. And if so, what about > sqrt of complex numbers? Does it give sqrt(i) = approx .707 + .707 i > or does it give both the + and - answer?
In the realm of complex analysis, all kinds of neat relationships exist. The first below is easily explainable via power series. In particular:
e^(a+b*i)=e^a*(cos b + i sin b)
If b>0, then ln(a+b*i)=ln(sqrt(a^2+b^2))+(arctan(a/b)+pi/2)i (and a similar mess if b<0)
And lots more.
For instance, arcsin(2) = pi/2 + i ln(sqrt(3)+2).
Dave Slomer AP Calculus and Computer Science Teacher Winton Woods HS, Cincinnati, OH 45240