What does a path integral measure in complex analysis?

[redshift]

What exactly does a path integral measure? Is it area between the ends/bounds of the line? Or is it the length of the line? Just started complex analysis and am comletely confused by this.

 

[shmoe]

What exactly does a path integral measure? Is it area between the ends/bounds of the line? Or is it the length of the line? Just started complex analysis and am comletely confused by this.

It's not the length, you can think of the dz as keeping track of direction and distance you move. The integral $\int_{\gamma}dz$ along a path $$z=\gamma(t),\ t\in\[a,b\]$$ is simply $\int_{a}^{b}\gamma '(t)dt=\gamma(b)-\gamma(a)$ so you actually get the distance and direction your end point is from your start point, not the total distance you've travelled. This integral doesn't even keep track of how the path made it from start to finish, nor how fast it went.

 

[wais]

A path integral is a mathematical tool used to calculate the value of a function along a specific path or curve. It is commonly used in calculus, physics, and other fields to solve problems involving motion, optimization, and probability.

The path integral measures the total contribution of a function over a given path or curve. It does not specifically measure the area or length of the line, but rather the overall effect of the function along that path. In other words, it takes into account the changes in the function as it moves along the path, rather than just the starting and ending points.

To better understand this, let's use an example. Imagine you are driving a car along a winding road. The path integral would measure the total distance traveled, taking into account any curves or turns in the road. It would not just measure the straight-line distance between the starting and ending points.

Similarly, in complex analysis, the path integral measures the total effect of a complex-valued function along a specific path in the complex plane. It takes into account the changes in the function as it moves along the path, rather than just the values at the starting and ending points.

I understand that this concept can be confusing, especially when just starting to learn about complex analysis. It may help to think of the path integral as a summation of small contributions along the path, with each contribution taking into account the changes in the function at that point. As the number of contributions becomes infinitely small, the path integral becomes more accurate.

I hope this explanation helps to clarify the concept of path integrals for you. Keep practicing and seeking out resources to deepen your understanding. Complex analysis can be challenging, but with patience and perseverance, you will grasp these concepts.