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In multivariable real calculus, if we want to calculate $\iint (x^2 + y^2) dx dy $ over a region R, this is a number that represents the volume under the surface given by $ (x^2 + y^2) $ and the xy- plane only above the region R. So, the geometric meaning of this operation is a volume.

If we wanna evalaute the intregral of a function $ f: \Bbb C \to \Bbb R $, over a path, for example, $f: \Bbb C\to \Bbb R$ with $ f(z) = Re(z^2) $ along a path I believe that, along that path, one is taking some alture and visualizing this we are making that operation to take the area between the path $\gamma $ in $\Bbb C$ and the "image path" of $\gamma$ in the "alture coordinate".

But if we have that $f(z) = z^2$ and we integrate alonge the same $\gamma$ path, what kind of geometric element is that?

And, what if we integrate the same function $f(z) = z^2$ but along the $\gamma $ path, indeed, what if we integrate $f(z) = z^2$ for example over all the closed unit circle region? What kind of geometric element is this?

Rodrigo Dias
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Juan Manuel Buchanan
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  • Cf. https://math.stackexchange.com/q/111368/96384, https://math.stackexchange.com/q/110334/96384, https://math.stackexchange.com/q/904493/96384, https://math.stackexchange.com/q/446724/96384 – Torsten Schoeneberg Apr 28 '24 at 01:33

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Let z=x+iy for $x,y \in \mathbb{R}$. We can generally write any function f(z) as f(x+iy)=u(x,y)+i*v(x,y) where u and v are both real valued functions. Then, we view the integral $\int_\gamma f(z)dz=\int_\gamma f(x+iy)dz=\int_\gamma (u(x,y)+i*v(x,y))dz=\int_\gamma u(x,y)dz+i\int_\gamma v(x,y)dz$. So, we see here that the integral is kind of like a normal path integral in $\mathbb{R}^2$, but instead of taking one, we take two and then sum them together. Hope that this helps.

Ben
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