Abstract Nonsense

The Exponential and Trigonometric Functions (Pt. II)

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

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This immediately tells us then that $\exp(x+iy)=e^x(\cos(y)+i\sin(y))$. This tells us quite a few thing:

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Theorem: Let $z\in\mathbb{C}$. Then, $|\exp(z)|=\exp(\text{Re}(z))$.

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Theorem: Let $w,z\in\mathbb{C}$. Then, $\exp(z)=\exp(w)$ if and only if $z-w\in2\pi i\mathbb{Z}$.

Proof: Since $\exp(z)=\exp(w)$ we have that $|\exp(z)|=|\exp(w)|$ and so the above theorem tells us (since the exponential function on $\mathbb{R}$ is injective) that $\text{Re}(z)=\text{Re}(w)$. We see then that $\cos(\text{Im}(z))+i\cos(\text{Im}(z))=\cos(\text{Im}(w))+i\cos(\text{Im}(w))$ and thus $\cos(\text{Im}(z))=\cos(\text{Im}(w))$ which tells us that $\text{Im}(z)-\text{Im}(w)\in 2\pi\mathbb{Z}$. Thus, evidently

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$z-w=(\text{Re}(z)-\text{Re}(w))+i(\text{Im}(z)-\text{Im}(w))\in 2\pi i\mathbb{Z}$

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The converse follows similarly. $\blacksquare$

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Corollary: The exponential function is periodic of period $2\pi i$.

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

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Theorem: The mapping $f:[0,1)\to\mathbb{S}^1$ given by $f(t)=\exp(2\pi i t)$ is a bijective continuous map.

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Remark: This is a simple example of why in general categories aren’t balanced. Indeed, the monos and epis in $\mathbf{Top}$ are the injective and surjective continuous maps. Thus, we see that $f$ is a bimorphism. That said, $f$ is not a homeomorphism (the isomorphism in $\mathbf{Top}$) otherwise there would exist a continuous bijection $\mathbb{S}^1\to [0,1)$ which is not true since $\mathbb{S}^1$ is compact and $[0,1)$ is not.

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Now, based on the above analysis we define the functions $\sin$ and $\cos$ mapping $\mathbb{C}\to\mathbb{C}$ by

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$\displaystyle \sin(z)=\frac{\exp(iz)-\exp(-iz)}{2i}=\sum_{n=0}^{\infty}\frac{(-1)^n z^{2n+1}}{(2n+1)!}$

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and

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$\displaystyle \cos(z)=\frac{\exp(iz)+\exp(-iz)}{2}=\sum_{n=0}^{\infty}\frac{(-1)^n z^{2n}}{(2n)!}$

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Since any linear combination of entire functions is entire it follows that both $\sin$ and $\cos$ are entire.

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References:

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[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.

May 5, 2012 -