Maclaurin series

[mooshupork34]

We have the following info about three functions A, B, and C for all values of T:

A(t))' = B(t)
(B(t))' = A(t)
(A(t))^2 - (B(t))^2 = 1
C(t) = B(t)/A(t)


From this, we can gather that the Maclaurin series for A(t) is 1 + t^2/2! + t^4/4! + t^6/6!... and the Maclaurin series for B(t) is t + t^3/3! + t^5/5! + t^7/7+...

a) What is the Maclaurin series for A(ix)?
b) What is the Maclaurin series for B(ix)?

 

[pka]

\(\displaystyle \L A(ix) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k x^{2k} }}{{\left( {2k} \right)!}}} \quad \& \quad B(ix) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k x^{2k + 1} }}{{\left( {2k + 1} \right)!}}} i\)

This leads to the Euler formula.