## The Exponential and Trigonometric Functions (Pt. II)

**Point of Post: **This is a continuation of this post.

This immediately tells us then that . This tells us quite a few thing:

**Theorem: ***Let . Then, .*

**Theorem: ***Let . Then, if and only if .*

**Proof: **Since we have that and so the above theorem tells us (since the exponential function on is injective) that . We see then that and thus which tells us that . Thus, evidently

The converse follows similarly.

**Corollary: ***The exponential function is periodic of period .*

We also note then that we get the following cool fact:

**Theorem: ***The mapping given by is a bijective continuous map.*

*Remark: *This is a simple example of why in general categories aren’t balanced. Indeed, the monos and epis in are the injective and surjective continuous maps. Thus, we see that is a bimorphism. That said, is not a homeomorphism (the isomorphism in ) otherwise there would exist a continuous bijection which is not true since is compact and is not.

Now, based on the above analysis we define the functions and mapping by

and

Since any linear combination of entire functions is entire it follows that both and are entire.

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**References:**

[1] Greene, Robert Everist, and Steven George Krantz. *Function Theory of One Complex Variable*. Providence: American Mathematical Society, 2006. Print.

[2] Conway, John B. *Functions of One Complex Variable I*. New York: Springer-Verlag, 1978. Print.

[3] Rudin, W. *Real and Complex Analysis.* New York,NY: McGraw-Hill, 1988. Print.

[4] Ahlfors, Lars V. *Complex Analysis; an Introduction to the Theory of Analytic Functions of One Complex Variable*. New York: McGraw-Hill, 1966. Print.

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