Some questions in Complex Analysis

In summary, the power function will be multivalued, and the problems of the Arg() function will propagate through the definitions and occur in z^a as well.
  • #1
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I'm not very clear of the problems below,so I may make some mistakes,if you point out them and explain to me,I'm reallly grateful.

1.If f(z) is an analytic function,why can we derivate it as a real function to get it's derivation?
I mean f'(z) should be [tex]f^' (z) = \frac{{\partial u}}{{\partial x}} + i\frac{{\partial v}}{{\partial x}}[/tex],we can get the derivation by this formula,but why can we just derivate it as a real function?For example,if [tex]f(z) = \log (z - a)[/tex],then it's derivation is
[tex]f'(z) = \frac{1}{{z - a}}[/tex]?


2.What on Earth is principle-valued branch?
Why (z)^(1/2) is multiple-valued?Why we may choose for [tex]\Omega [/tex] the complement of the negative real axis z<=0 then it is a single-valued function?
I'm really confused of it.And why once the continuity is established the analyticity follows by derivation of the inverse function?
 
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  • #2
1. That's the result, it's not a definition. You start with the definition of the derivative and then you get that result. It's not surprising though, since the definition of derivative in complex analysis is the same as in real analysis, except you put dz instead of dx. But the analyticity constraint makes the derivative independent of the direction you take dz to approach zero, therefore you expect the result to be like in real analysis. But to conclude, this is the result of the derivative identity (which a result of the analyticty constraint), try it yourself.

2. The complex power is defined in this way:

[tex]z^{w}=e^{log(z^{w})}=e^{wlog(z)}[/tex]

Because we know how to deal with exponents and logarithms, but the idea of a complex (or even a real irrational) power is obscure to us.

Since the log itself is a multivalued function (since it contains the arg() function as the imaginary part), the power function will be multivalued itself.

The principle branch of the log(z) function, also denoted Log(z) will also define the principle branch of [tex]f(z)=z^{\alpha}[/tex]

And since all of the problems of Arg() function arise in the negative real axis, these problems will propagate through the definitions and occur in z^a as well.
 
  • #3
Hi elibj123 ,thanks very much for your explanation.

One more question,why all of the problems of Arg() function arise in the negative real axis? Why not the positive real axis?
 

1. What is Complex Analysis?

Complex Analysis is a branch of mathematics that deals with the study of complex numbers and functions. It is a combination of complex algebra and calculus, and it is used to understand the properties and behavior of functions that involve complex numbers.

2. What are complex numbers?

Complex numbers are numbers that have both a real and an imaginary part, and can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Complex numbers are useful in representing quantities that involve both real and imaginary components, such as in electrical engineering and physics.

3. What is the difference between real analysis and complex analysis?

Real analysis deals with the study of real numbers and functions, while complex analysis focuses on complex numbers and functions. Real analysis is limited to one-dimensional functions, while complex analysis can be extended to higher dimensions. Additionally, complex analysis has its own set of tools and techniques that are not applicable in real analysis.

4. What are some applications of Complex Analysis?

Complex Analysis has numerous applications in mathematics, physics, engineering, and other fields. It is used in the study of fluid dynamics, electromagnetism, signal processing, and quantum mechanics. It is also used in the development of mathematical models and algorithms for solving complex problems.

5. What are some common theorems in Complex Analysis?

Some commonly used theorems in Complex Analysis include the Cauchy-Riemann equations, Cauchy's Integral Theorem, the Fundamental Theorem of Algebra, and the Maximum Modulus Principle. These theorems are used to study the behavior of complex functions and to solve problems in various areas of mathematics and science.

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