I am currently learning complex analysis and I am trying to interpret complex line integral in terms of physics concepts, such as force and energy. In particular, I want to know if the real part of a complex line integral represents the work done required to move along the curve.
This is because for a curve $\gamma (t)$, and a function $f=u+iv$, we have:
$$\int_{\gamma }f(z)dz=\int_{\gamma}(u,-v)\cdot dr +i\int_{\gamma}(u,-v)\cdot N dt$$
If we define the vector field $g=(u,-v)$, we can rewrite it as:
$$\int_{\gamma }f(z)dz=\int_{\gamma} g \cdot dr +i\int_{\gamma} g \cdot N dt$$
So we have:
$$ \textbf{Re}\left [\int_{\gamma }f(z)dz \right ]=\int_{\gamma } g\cdot dr $$
To be specific, may I ask the following questions:
Does $\int_{\gamma} g \cdot dr$ represent the work done required to move along the curve, given that a force $F=(u,-v)$ is acting on the moving point?
My lecture notes says that $\int_{\gamma} g \cdot dr$ represents the circulation of $g$, is circulation same as work done along a closed curve? I am confused since my lecture notes didn't say the curve $\gamma$ is closed.
Any help will be appreciated!
In some contexts, you can interpretate it as the work done by a given force, but this is only valid if you consider your function $f(z)$ as a Force Vector Field. Is well known that a complex function $f(z)$ can represent physical situations, like Forces Fields, ElectroMagnetic Fields, Fluids, weather situations, etc. It all depends in the interpretation that you give to $f(z)$ (for example, if f(z) doesn't represent a force field, then $\int_\gamma g \cdot dr$ can be interpretate as the work done)
i'm not totally sure, but the circulation is a more mathematical definition or quantity than a physical one. And it's not necessary that the curve $\gamma$ has to be a close one.
Good luck in your futures learnings! I highly recommend this video to gain some intuition in the complex plane: https://www.youtube.com/watch?v=EyBDtUtyshk&t=1936s&ab_channel=Mathemaniac It helped me in it's moment!