5) Proof of Euler’s formula using power series expansions.
I forgot that this is only insightful if you already realized the following:
(3+4i) =
(r, angel) =
(sqrt((3^2)+(4^2)), arctan(4/3)) =
(5, 53.1301) =
r(cos(x)+isin(x)) =
5(cos(arctan(4/3))+isin(arctan(4/3))) =
5e^(arctan(4/3)*i) =
e^ln(5)*e^(arctan(4/3)i) =
e^(ln(5)+arctan(4/3)i)
Only the cartesian, polar and cos-sin forms were obvious to me, and I was still able to make sense of the Taylor series proof.
I forgot that this is only insightful if you already realized the following:
(3+4i) =
(r, angel) =
(sqrt((3^2)+(4^2)), arctan(4/3)) =
(5, 53.1301) =
r(cos(x)+isin(x)) =
5(cos(arctan(4/3))+isin(arctan(4/3))) =
5e^(arctan(4/3)*i) =
e^ln(5)*e^(arctan(4/3)i) =
e^(ln(5)+arctan(4/3)i)
Only the cartesian, polar and cos-sin forms were obvious to me, and I was still able to make sense of the Taylor series proof.