Abstract Nonsense

Crushing one theorem at a time

Complex Differentiable and Holmorphic Functions (Pt. II)


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

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Complex Differentiablility 

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Let \Omega\subseteq\mathbb{C} be open (we are obviously endowing \mathbb{C}\approx\mathbb{R}^2 with the usual topology). We say that a function f:\Omega\to\mathbb{C} is differentiable at z_0\in\Omega if there exists a \mathbb{C}-linear function A:\mathbb{C}\to\mathbb{C} such that the limit

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\displaystyle \lim_{z\to z_0}\frac{f(z)-f(z_0)-A(z-z_0)}{z-z_0}=0

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We call A the complex total derivative of f at z_0 and denote it \widetilde{D}_f((z_0). Of course, this definition is pure pedantry since any \mathbb{C}-linear function \mathbb{C}\to\mathbb{C} is of the form z\mapsto \alpha z for some complex number \alpha. Thus, we know that \widetilde{D}_f(z_0), if it exists, is secretly just multiplication by some constant. We denote this constant f'(z_0) and call it the derivative of f at z_0. We see then that f being differentiable at z_0 is equivalent to the statement that

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\displaystyle \lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}

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exists, and when it does, it is equal to f'(z_0).

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If f is differentiable at every point of \Omega we say it’s differentiable on \Omega. Note then that we get a function f':\Omega\to\mathbb{C} defined by z_0\mapsto f'(z_0). We say that f is continuously differentiable if f' is continuous on \Omega. We say that f is twice differentiable if f' is differentiable on \Omega, and we define n-times differentiable similarly.

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The first key theorem about differentiable functions is the following:

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Theorem: Let \Omega\subseteq\mathbb{C} be open. Then, if f:\Omega\to\mathbb{C} is such that f is complex differentiable at z_0 then it is real differentiable at z_0=x_0+iy_0=(x_0,y_0) and \widetilde{D}_f(z)=D_f(z) (i.e. the complex total derivative is equal to the real total derivative).

Proof: Identifying \mathbb{C}\cong\mathbb{R}^2 as \mathbb{R}-vector spaces we see that \widetilde{D}_f(z) is naturally an \mathbb{R}-map \mathbb{R}^2\to\mathbb{R}^2 which evidently satisfies

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\displaystyle \lim_{(x,y)\to (x_0,y_0)}\frac{\|f(x,y)-f(x_0,y_0)-\widetilde{D}_f(x-x_0,y-y_0)\|}{\|(x,y)-(x_0,y_0)\|}=0

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from where the conclusion follows by the definition of being real differentiable and being a total derivative. \blacksquare

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This allows us to conclude about the complex derivative all the things that were true for real differentiable functions \mathbb{R}^2\to\mathbb{R}^2. For example:

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Corollary(Chain Rule): Let \Omega,\Omega'\subseteq\mathbb{C} be open and suppose that f:\Omega\to\Omega' and g:\Omega'\to\mathbb{C} are functions such that f is differentiable at z_0\in\Omega and g is differentiable at f(z_0)\in\Omega. Then, g\circ f:\Omega\to\mathbb{C} is differentiable at z_0 and (g\circ f)'(z_0)=g'(f(z_0))\cdot f'(z_0).

Proof: By the chain rule for total derivatives and the fact that \widetilde{D}=D (complex total derivative equals total derivative) we know that \widetilde{D}_{g\circ f}(z_0)=\widetilde{D}_g(f(z_0))\circ \widetilde{D}_f(z_0), but this is clearly just the linear map z\mapsto g'(f(z_0))f'(z_0) from where the conclusion follows. \blacksquare

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Using the same idea we also can conclude the following:

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Corollary: Let \Omega\subseteq\mathbb{C} be open and f,g:\Omega\to\mathbb{C}. Suppose that f,g are differentiable at z_0\in\Omega then:

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\begin{aligned}&\mathbf{(1)}\quad f+g\text{ is differentiable at }z_0\text{ and }(f+g)'(z_0)=f'(z_0)+g'(z_0)\\ &\mathbf{(2)}\quad fg\text{ is differentiable at }z_0\text{ and }(fg)'(z_0)=f'(z_0)g(z_0)+f(z_0)g'(z_0)\\ \displaystyle &\mathbf{(3)}\quad \frac{f}{g}\text{ is differentiable at }z_0\text{ and }\left(\frac{f}{g}\right)'(z_0)=\frac{g(z_0)f'(z_0)-f(z_0)g'(z_0)}{g(z_0)^2}\end{aligned}

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where we must assume that g(z_0)\ne 0 for the last one.

Proof: The first and last one we have proven before (here and here respectively) and the second can be proved thinking of multiplication as a bilinear map and applying this result. \blacksquare

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We also know this fact:

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Theorem: Let \Omega\subseteq\mathbb{C} be open and connected. Then, if f'=g' as functions on \Omega then f=g+c for some constant c. In particular, if f'=0 then f is constant.

Proof: Apply the case for real functions. \blacksquare

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Since we know that real differentiable functions are continuous we may also conclude this:

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Theorem: Let \Omega\subseteq\mathbb{C} be open. If f:\Omega\to\mathbb{C} is differentiable at z_0\in\Omega then it is continuous at z_0.

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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 1, 2012 - Posted by | Complex Analysis, Uncategorized | , , , , , , , , , , , , ,

2 Comments »

  1. […] Complex Differentiable and Holmorphic Functions (Pt. III) Point of Post: This is a continuation of this post. […]

    Pingback by Complex Differentiable and Holmorphic Functions (Pt. III) « Abstract Nonsense | May 1, 2012 | Reply

  2. […] we’d have that is entire (note we can divide since is never zero). That said, using the division rule we see […]

    Pingback by The Exponential and Trigonometric Functions « Abstract Nonsense | May 5, 2012 | Reply


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