What is the basic assumption behind the Cauchy-Riemann equation?

[iVenky]

Recently I read about Cauchy-Riemann equations and I got a doubt in that.

I can understand the derivation of it but I couldn't understand the basic assumption with which you derive that.

Why should the limit give the same value for both real and for imaginary axis?

I hope you can understand the question. If you can't understand the question just reply.

Thanks a lot

 

[Bacle2]

It comes down to the fact that for the limit to exist, it should exist from every direction

you approach a point. In particular, it must exist as you approach along the real

axis as well as when you approach along the imaginary axis, and the two limits

must be equal.

 

[iVenky]

That point is valid if you are finding just the limit as in

$\lim_{x \to a} f(x,b)$ to be same as
$\lim_{y \to b} f(a,y)$

But it is not necessary that the differentiation with respect to real and imaginary axis had to be same, right? I mean it's like partial derivative where derivative with respect to one axes (real in this case) need not be same as the derivative with respect to another axes (imaginary axes).

thanks a lot for your help.

 

[Bacle2]

That point is valid if you are finding just the limit as in

$\lim_{x \to a} f(x,b)$ to be same as
$\lim_{y \to b} f(a,y)$

But it is not necessary that the differentiation with respect to real and imaginary axis had to be same, right? I mean it's like partial derivative where derivative with respect to one axes (real in this case) need not be same as the derivative with respect to another axes (imaginary axes).

thanks a lot for your help.

But you are finding a limit; the derivative is a limit , the quotient limit

[f(z+zo)-f(z)]/(z-zo) as z→zo . But z can approach zo along _every possible

complex direction. Then f'(z) exists when this limit exist,so that the limit must

exist along any direction along which you approximate zo, and for the limit to

exist, it must be the same no-matter how you approximate zo. In particular,

(re Cauchy-Riemann) , the approximation along the x-axis and the y-axis must

exist, and must agree with each other.

 

[blue_raver22]

.

The basic assumption behind the Cauchy-Riemann equation is the concept of analyticity. This means that a function is differentiable at every point in a given region. In other words, the function is smooth and has no abrupt changes or corners. This assumption allows us to apply the rules of calculus to complex functions, which are functions that have both real and imaginary components.

The Cauchy-Riemann equation is derived from the idea that a complex function can be expressed as a combination of two real-valued functions: one for the real component and one for the imaginary component. By applying the rules of calculus to these two functions, we can derive the Cauchy-Riemann equation, which states that the partial derivatives of the real and imaginary components are related to each other in a specific way.

The reason why the limit must give the same value for both the real and imaginary axis is because the Cauchy-Riemann equation is based on the assumption of analyticity. If the function is not analytic, then the Cauchy-Riemann equation does not hold and the limit will not give the same value for both axes. However, in most cases, we are interested in functions that are analytic, as they have useful and interesting properties.

I hope this helps to clarify the basic assumption behind the Cauchy-Riemann equation. If you have further questions, please feel free to reply and I will do my best to address them. Thank you for your interest in this topic.