Abstract Nonsense

Crushing one theorem at a time

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.

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May 5, 2012 - Posted by | Complex Analysis | , , ,

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