Is the Integral Zero for Closed Paths in Complex Analysis?

In summary, the conversation discusses how to show that an integral is equal to zero and whether the condition in the premise is on the absolute value of a function or the function itself. It is mentioned that the integral must be equal to zero if it is less than or equal to zero, and that being a closed path may also contribute to the integral being equal to zero.
  • #1
Gwinterz
27
0
Hey, I have been stuck on this question for a while:

WClcaAN.jpg

I have tried to follow the hint, but I am not sure where to go next to get the result.



Have I started correctly? I am not sure how to show that the integral is zero.

If I can show it is less than zero, I also don't see how that shows it is always zero.

Thanks in advance for any help.
 

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  • #2
The condition in the premise is on ##\left|f(z)\right|##, not ##f(z)##. If you have in fact shown that ##\left|f(z)\right| \leq 0##, then it must be equal to 0 since it can't be negative.
 
  • #3
Gwinterz said:
Hey, I have been stuck on this question for a while:

View attachment 232707
I have tried to follow the hint, but I am not sure where to go next to get the result.



Have I started correctly? I am not sure how to show that the integral is zero.

If I can show it is less than zero, I also don't see how that shows it is always zero.

Thanks in advance for any help.

Something cannot be less than zero and equal to zero at the same time. However, since you have non-strict inequalities "##\leq##" there is a chance you can show the thing is ##\leq 0##. Then (being a norm in the complex plane) it must also be ##\geq 0##, hence must ##= 0.##
 
  • #4
Thanks guys that makes sense.

Is the integral equal to zero because its a closed path?
 

Related to Is the Integral Zero for Closed Paths in Complex Analysis?

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and functions. It involves the analysis of functions with complex variables and the properties and behavior of these functions.

2. What are complex numbers?

Complex numbers are numbers that have both a real and imaginary part. They are written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

3. What are the applications of complex analysis?

Complex analysis has many applications in mathematics, physics, engineering, and other fields. It is used in the study of differential equations, signal processing, fluid dynamics, and quantum mechanics, among others.

4. What are some key concepts in complex analysis?

Some key concepts in complex analysis include analytic functions, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem. Other important concepts include Laurent series, residue theory, and conformal mapping.

5. How is complex analysis different from real analysis?

Complex analysis deals with functions with complex variables, while real analysis deals with functions with real variables. In complex analysis, functions are defined in terms of complex numbers and their properties, while in real analysis, functions are defined in terms of real numbers. Additionally, complex analysis involves studying functions in the complex plane, while real analysis involves studying functions on the real number line.

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