- #1
GGGGc
- TL;DR Summary
- Why does Gauss mean value theorem means f(z
0) is equal to the average of f(z) around the rim of any circle in R centre at z0? The denominator is 2pi instead of 2pi * radius which is the circumference of the circle.
The short answer is because we are integrating over the angle, not over the circumference.GGGGc said:TL;DR Summary: Why does Gauss mean value theorem means f(z
0) is equal to the average of f(z) around the rim of any circle in R centre at z0? The denominator is 2pi instead of 2pi * radius which is the circumference of the circle.
View attachment 335776
Thanks for replying. Does that mean the average of f(z) is equal to f(z)/2pi since we are integrating it from 0-2pi?topsquark said:The short answer is because we are integrating over the angle, not over the circumference.
The slightly longer answer is that we are concerned primarily with the integral over the phases of the complex numbers surrounding z, not so much with r which, as you may note, is irrelevant due to Cauchy's integral formula.
-Dan
No, the integral of a function f(z) (analytic over a circle of radius a about the point z) over a circle in the complex plane of radius r (r < a) about the point z is ##2 \pi f(z)##. It is an "average" in the sense that we are looking at the sum over the values of a region and that the points in the region of integration that aren't equal to z are essentially meaningless, except for a constant normalizing factor of ##2 \pi##.GGGGc said:Thanks for replying. Does that mean the average of f(z) is equal to f(z)/2pi since we are integrating it from 0-2pi?
Many thanks for the explanation! I understand now. Thank youtopsquark said:No, the integral of a function f(z) (analytic over a circle of radius a about the point z) over a circle in the complex plane of radius r (r < a) about the point z is ##2 \pi f(z)##. It is an "average" in the sense that we are looking at the sum over the values of a region and that the points in the region of integration that aren't equal to z are essentially meaningless, except for a constant normalizing factor of ##2 \pi##.
Putting it a different way, to find the average of a real function:
##\displaystyle \overline{f} = \dfrac{1}{b - a} \int_a^b f(x) \, dx##
and the "average" of a complex function:
##\displaystyle \overline{f}(z) = \dfrac{1}{2 \pi} \int f(z + r e^{i \theta}) \, d \theta##
There are similarities to an average, so you can actually call it that (hey, that's what the formula is called!) but it isn't quite the same as an average in the sense of the mean value theorem.
-Dan
Gauss' mean value theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the derivative of the function is equal to the average rate of change of the function over the closed interval.
Gauss' mean value theorem is used in calculus to prove the existence of points where the derivative of a function is equal to the average rate of change of the function over a given interval. This theorem is often used to simplify calculations and prove important results in calculus.
Gauss' mean value theorem is significant because it provides a powerful tool for analyzing the behavior of functions over intervals. By guaranteeing the existence of certain points with specific properties, this theorem helps mathematicians and scientists better understand the behavior of functions and make predictions based on their properties.
Gauss' mean value theorem can only be applied to functions that are continuous on a closed interval and differentiable on the open interval. If a function does not meet these criteria, then the theorem cannot be used to prove the existence of points with specific properties.
Gauss' mean value theorem is a generalization of the mean value theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the derivative of the function is equal to the slope of the secant line passing through the endpoints of the interval. Gauss' mean value theorem extends this concept by considering the average rate of change of the function over the closed interval.