Expectation value of an angular momentum with a complex exponent

[isochore]

Homework Statement

Find $\langle l, m \vert \exp((a+ bi)L_z) \vert l, m \rangle$

Relevant Equations

$L_z \vert l, m \rangle = \hbar m \vert l, m \rangle$
$[L_x, L_y] = i \hbar L_z$

I am struggling to figure out how to calculate the expectation value because I am finding it hard to do something with the exponential. I tried using Euler's formula and some commutator relations, but I am always left with some term like $\exp(L_z)$ that I am not sure how to get rid of.

 

[Dr Transport]

Taylor series for the exponent...

 

[isochore]

Taylor series for the exponent...

Ahh, not sure why I did not think of that...

So you can expand starting with the original expression $\langle l, m \vert \exp((a+bi)L_z) \vert l, m \rangle$, then since $L_z = \hbar m \vert l, m \rangle$ you can simply compress back into something like $\exp (\hbar m (a+bi))$?

 

[Dr Transport]

Looks correct...