Limits of Complex Functions .... Zill & Shanahan, Theorem 3.1.1/ A1

In summary, Peter is trying to figure out if Theorem 3.1.1 (A1) can be generalized to a higher-dimensional space. He is also looking into the definition of limits and what the difference is between limits in ℝ2 and ℂ.
  • #1
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I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ...

I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ...

The statement of Theorem 3.1.1 (A1) reads as follows:

View attachment 9225
In the proof of Theorem 3.1.1 (A1) [see below] we read the following:

" ... ... On the other hand, with the identifications \(\displaystyle f(z) = u(x,y) + i v(x,y)\) and \(\displaystyle L = u_0 + i v_0\), the triangle inequality gives \(\displaystyle \mid f(z) - L \mid \ \leq \ \mid u(x,y) - u_0 \mid + \mid v(x,y) - v_0 \mid\) ... ... "
My question is as follows:

How exactly do we apply the triangle inequality to get \(\displaystyle \mid f(z) - L \mid \ \leq \ \mid u(x,y) - u_0 \mid + \mid v(x,y) - v_0 \mid\) ... ... ? Note: Zill and Shanahan give the triangle inequality as

\(\displaystyle \mid z_1 + z_2 \mid \ \leq \ \mid z_1 \mid + \mid z_2 \mid\)

My thoughts are as follows:

\(\displaystyle \mid f(z) - L \mid \ = \ \mid u(x,y) + i v(x,y) - (u_0 + i v_0 ) \mid \ = \ \mid ( u(x,y) - u_0 ) + i ( v(x,y) - v_0 ) \mid\)so in triangle inequality put

\(\displaystyle z_1 = u(x,y) - u_0 + i.0\)

and

\(\displaystyle z_2 = 0 + i ( v(x,y) - v_0 \))and apply triangle inequality ...Is that correct?Hope someone can help ...

Peter==================================================================================The statement and proof of Theorem 3.1.1 (given in Appendix 1 where the theorem is called Theorem A.1) reads as follows:
View attachment 9226
View attachment 9227Hope that helps ...

Peter
 

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  • #2
I'm going to suggest 2 things to streamline this. The main question is: what if, instead of $\mathbb C$ this was $\mathbb R^2$?

First, in general this is a good questions to ask as $\mathbb C$ in some sense is $\mathbb R^2$ with some 'tiny refinements' (i.e. geometrically pleasant field operations). One of the main points of complex analysis is that these 'tiny refinements' add a huge amount of structure and while definitions formally look about the same e.g. for differentiability on an open set in $\mathbb R^2$ vs $\mathbb C$, all kinds of things can be differentiable for the former that aren't for the latter -- and as a result only 'nice' functions are called holomorphic. So it's always a good question to have in the back of your mind and to try to make these comparisons.

Second, check on the definition of the limit in $\mathbb C$ vs that in $\mathbb R^2$. Consider $\mathbf z \in \mathbb R^2$ and
$f: \mathbb R^2 \to \mathbb R^2$. You'll see that $u(x,y) = z_1$ and $i v(x,y) = z_2$, i.e. they are isolating real and imaginary parts just like in the vector space interpretation of $\mathbb C$. Now apply results from your most recent thread about convergence in $\mathbb R^d$ iff there is component-wise convergence, selecting $d = 2$.
 
  • #3
Yeah this shall be helpful, as one of the main points of complex analysis is that these 'tiny refinements' add a huge amount of structure and while definitions formally look about the same e.g. for differentiability on an open set in ℝ2 vs ℂ, all kinds of things can be differentiable for the former that aren't for the latter -- and as a result only 'nice' functions are called holomorphic. So it's always a good question to have in the back of your mind and to try to make these comparisons.
 
  • #4
frapps11 said:
Yeah this shall be helpful, as one of the main points of complex analysis is that these 'tiny refinements' add a huge amount of structure and while definitions formally look about the same e.g. for differentiability on an open set in ℝ2 vs ℂ, all kinds of things can be differentiable for the former that aren't for the latter -- and as a result only 'nice' functions are called holomorphic. So it's always a good question to have in the back of your mind and to try to make these comparisons.
Thanks to steep and frapps11 for most helpful posts ...

I am still reflecting on what you have written ...

Thanks again!

Peter
 

FAQ: Limits of Complex Functions .... Zill & Shanahan, Theorem 3.1.1/ A1

What is the significance of Theorem 3.1.1/A1 in the study of complex functions?

Theorem 3.1.1/A1, also known as the Cauchy-Riemann equations, is a fundamental theorem in the study of complex functions. It provides necessary and sufficient conditions for a complex function to be differentiable at a point, and is essential in understanding the behavior of complex functions.

Can you explain the Cauchy-Riemann equations in simple terms?

The Cauchy-Riemann equations state that a complex function is differentiable at a point if and only if its partial derivatives with respect to the real and imaginary parts of the input variable exist and satisfy a certain relationship. This relationship is known as the Cauchy-Riemann conditions.

How are the Cauchy-Riemann equations related to the concept of analyticity?

A complex function that satisfies the Cauchy-Riemann equations is said to be analytic at a point. This means that the function can be represented by a convergent power series in a neighborhood of that point, and has derivatives of all orders at that point. In other words, the function is infinitely differentiable at that point.

Are there any practical applications of Theorem 3.1.1/A1?

The Cauchy-Riemann equations have numerous applications in mathematics and physics. They are used in the study of fluid dynamics, electromagnetism, and quantum mechanics, among others. They also play a crucial role in the development of complex analysis, which has wide-ranging applications in engineering and technology.

How can Theorem 3.1.1/A1 be extended to higher dimensions?

The Cauchy-Riemann equations can be extended to higher dimensions through the use of the Cauchy-Riemann equations for multivariable functions. These equations involve partial derivatives with respect to multiple variables and provide conditions for a function to be differentiable in higher dimensions. They are essential in the study of vector calculus and differential geometry.

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