The discussion revolves around the importance of the Cauchy-Riemann relations in the context of complex differentiability. Participants explore the implications of these relations, their definitions, and their connections to the concept of analytic functions, as well as the nature of the variables involved.
Participants express differing opinions on the interpretation of the Cauchy-Riemann relations and the nature of \(z\) and \(\bar{z}\). There is no consensus on whether they should be treated as independent variables or the implications of the Cauchy-Riemann equations as definitions versus necessary conditions for differentiability.
Some arguments rely on specific interpretations of complex differentiability and the definitions of analytic functions, which may vary among different texts. The discussion also highlights the complexity of functions that may not appear to depend on \(\bar{z}\) but are still influenced by it.
<br /> <br /> What exactly do you mean? Do you want to take the derivative of the complex part?<br /> <br /> The Cauch-Riemann equations can be used to see the following: A holomorphic function f:\mathbb{C}\rightarrow\mathbb{C} is defined by its real part, plus some constant function.<br /> <br /> <u>Proof:</u> Let's look at two holomorphic functions f,g. Let Re(f)=Re(g), then look at the new function h defined by h:=f-g. From the properties of complex differentiation we know that this function h also is complex differentiable and because of Re(f)=Re(g), we know that Re(h)=0. Let h=u+iv for some real functions u,v (You should know that this can be done, otherwise consult a complex analysis book like Rudin). In our special case we know that u=0, and therefore any derivative of u is also equal to 0. The Cauchy-Riemann equations now state that 0=\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} and 0=\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}. That's it. Both partial derivatives of v are equal to 0, so the imaginary part of h is constant, and thus the difference of f and g is constant.<br /> <br /> Is it clear?Wiemster said:The cauchy Riemann relations can be written:
\frac{\partial f}{\partial \bar{z}}=0
Is there an 'easy to see reason' why a function should not depend on the independent variable \bar{z}[/tex] to be differentiable?
matt grime said:It is a definition, Weimster. I'm not sure what you're asking. It is the definition of analytic.
cliowa said:What exactly do you mean? Do you want to take the derivative of the complex part?
The Cauch-Riemann equations can be used to see the following: A holomorphic function f:\mathbb{C}\rightarrow\mathbb{C} is defined by its real part, plus some constant function.
Proof: Let's look at two holomorphic functions f,g. Let Re(f)=Re(g), then look at the new function h defined by h:=f-g. From the properties of complex differentiation we know that this function h also is complex differentiable and because of Re(f)=Re(g), we know that Re(h)=0. Let h=u+iv for some real functions u,v (You should know that this can be done, otherwise consult a complex analysis book like Rudin). In our special case we know that u=0, and therefore any derivative of u is also equal to 0. The Cauchy-Riemann equations now state that 0=\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} and 0=\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}. That's it. Both partial derivatives of v are equal to 0, so the imaginary part of h is constant, and thus the difference of f and g is constant.
Is it clear?
matt grime said:I wish you wouldn't say 'indepedent of z bar'. That isn't how to think of it. z and z bar are not independent variables - if I give you z you know what z bar is.