Understanding Differentiability and Continuity in Complex Analysis

In summary, the two books discuss the proofs that for a function of a complex variable, differentiability implies continuity. While the proofs presented by John B. Conway and John H. Mathews and Russel W. Howell (M&H) are slightly different, they are both correct and equivalent. The main difference is that Conway uses modulus/norm signs in his proof, while M&H do not. Some may argue that Conway's inclusion of these signs is unnecessary, but it ultimately does not affect the validity of the proof.
  • #1
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I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ...

The two books are as follows:

"Functions of a Complex Variable I" (Second Edition) ... by John B. Conway

"Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... Conway's proof that for a function of a complex variable, differentiability implies continuity ... reads as follows:
View attachment 9258
Mathews and Howell's proof that for a function of a complex variable, differentiability implies continuity ... reads as follows:
View attachment 9259
Now, as can be seen in the above proofs, Conway uses modulus/norm signs around the expressions in the proof while Mathews and Howell do not ...Can someone explain the differences ... are both correct ... ?

Surely the Conway proof is more valid as the proof involves limits which involve ideas like "close to" which need modulus/norms ...Hope someone can clarify this issue ...

Peter
 

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  • #2
They are equivalent. If [tex]\lim_{x\to a} f(x)= b[/tex] then [tex]\lim_{x\to a} |f(x)|= |b|[/tex] and if b= 0 the converse is also true.
 
  • #3
HallsofIvy said:
They are equivalent. If [tex]\lim_{x\to a} f(x)= b[/tex] then [tex]\lim_{x\to a} |f(x)|= |b|[/tex] and if b= 0 the converse is also true.

Thanks for the help HallsofIvy ...

But ... it leaves me thinking that Conway made a pointless elaboration of his proof as modulus/norm signs were unnecessary ... indeed, I have no idea why he included them ...

Peter
 

What is the definition of differentiability in complex analysis?

In complex analysis, differentiability refers to the property of a complex-valued function of having a derivative at a point. This means that the function is smooth and has a well-defined slope at that point.

How is differentiability related to continuity in complex analysis?

Differentiability and continuity are closely related in complex analysis. A function is differentiable at a point if and only if it is continuous at that point. This means that if a function is not continuous at a point, it cannot be differentiable at that point.

What is the Cauchy-Riemann equations and how do they relate to differentiability?

The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a complex-valued function to be differentiable at a point. These equations relate the partial derivatives of the real and imaginary parts of a function, and if they are satisfied, the function is differentiable at that point.

Can a function be differentiable but not analytic?

Yes, a function can be differentiable at a point but not analytic in a neighborhood of that point. This means that the function does not have a power series expansion around that point, and therefore cannot be represented as a complex polynomial.

What are the implications of a function being analytic in a region?

If a function is analytic in a region, it means that it is differentiable at every point in that region, and can be represented as a complex polynomial. This has important implications in complex analysis, as it allows for the use of powerful tools such as Cauchy's integral theorem and Cauchy's integral formula.

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