Complex Derivative .... Remark in Apostol, Section 16.1 .... ....

In summary, Peter is reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) and is focused on Chapter 16: Cauchy's Theorem and the Residue Calculus. He needs help understanding a remark made by Apostol in Section 16.1, which asks for a demonstration of how a complex function f can have a derivative at 0 but at no other point in the complex plane. Peter realizes that the Cauchy-Riemann equations are only satisfied at (0,0), making it the only possible point where the derivative of f can exist. He also notes that since the functions u and v have continuous first-order partial derivatives, f must have a derivative at (0,0
  • #1
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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...

I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ...

I need help in order to fully understand a remark of Apostol in Section 16.1 ...

The particular remark reads as follows:

View attachment 9279Could someone please demonstrate (in some detail) how it is the case that the complex function \(\displaystyle f\) has a derivative at \(\displaystyle 0\) but at no other point of \(\displaystyle \mathbb{C}\) ... ...Help will be much appreciated ...

Peter
 

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  • #2
Hi Peter,

With $z=x+iy$, you have $f(z)=u + iv$, with $u=x^2+y^2$ and $v=0$.

What do the Cauchy-Riemann equations tell you ?
 
  • #3
castor28 said:
Hi Peter,

With $z=x+iy$, you have $f(z)=u + iv$, with $u=x^2+y^2$ and $v=0$.

What do the Cauchy-Riemann equations tell you ?
Oh! Indeed ... Cauchy-Riemann equations are only satisfied at (0,0) ... therefore the only possible point where the derivative of f can exist is (0,0) ... and, given that the functions u and v ere continuous and have continuous first-order partial derivatives then f has a derivative at (0,0) ...

Thanks fir the help ... it is much appreciated ...

Peter
 

FAQ: Complex Derivative .... Remark in Apostol, Section 16.1 .... ....

What is a complex derivative?

A complex derivative is a mathematical concept that extends the idea of a derivative to complex-valued functions. It measures the rate of change of a complex function with respect to its complex input variable.

Why is the complex derivative important?

The complex derivative is important because it allows us to extend the tools and techniques of calculus to complex functions, which are often used in physics, engineering, and other fields.

What is the difference between a complex derivative and a real derivative?

The main difference between a complex derivative and a real derivative is that a complex derivative takes into account both the real and imaginary parts of a function, while a real derivative only considers the real part. Additionally, the complex derivative can have a direction, unlike the real derivative which only has a single value.

How is the complex derivative calculated?

The complex derivative is calculated using the limit definition of a derivative, similar to the real derivative. However, in the complex case, the limit is taken along any path in the complex plane, rather than just along the real axis.

What is the remark in Apostol, Section 16.1 about complex derivatives?

In Apostol, Section 16.1, the remark discusses the Cauchy-Riemann equations, which are necessary conditions for a function to be complex differentiable. It also mentions that a function can be differentiable at a point without being analytic (having a complex derivative at every point in a region).

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