Exponential in basis

Definition 4.1.1   If , with real , we define:

.

Proposition 4.1.2       

(i) The exponential is defined on and .
(ii) The exponential is an entire function.
(iii) .
(iv) .
(v) .

Proof.     

(i) Let . Then . As for any real number , , we have .
(ii) As above, let . Then we have:

It can be easily shown that the functions and verify the Cauchy-Riemann equations over the whole plane. As the plane is open, the exponential function is differentiable at every point of an open set, whence analytic at every point. This means that the function is an entire function.
(iii) Denote and , with . Then we have:

(iv) Proceed as for (iii).
(v) For any , we have:

Example 4.1.3   Let and . Then:

Example 4.1.4   Solve the equation in .

Let , where are real numbers. We have:

As , for any , we have , i.e. . We consider now two cases:

(i) If , with , we have . The first equation has one solution, given by .
(ii) If , with , we have . The first equation implies now that , and has no solution.

We conclude: the solution set of the given equation in is .

Example 4.1.5   Solve the equation in . Let , where are real numbers. We have:

As , for any , we have , i.e. . We consider now two cases:

(i) If , for , we have . The second equation implies , i.e. .
(ii) If , for , we have .The second equation implies ,which has no real solution.

We conclude: the solution set of the given equation in is .