Find e^cos(2+3i) as a complex number expressed in cartesian form.

The complex number has Cartesian form

.

Step-by-step explanation:

First, we need to recall the definition of when is a complex number:

.

Then,

. (I)

Now, recall the definition of the complex exponential:

.

So,

(we use that .

Thus,

Now we group conveniently in the above expression:

.

Now, substituting this equality in (I) we get

.

Thus,

.