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There are two approaches to defining $\int_{\gamma}f$.

The first is to build up the theory of complex Riemann sums by mimicking the real case.
The second is to define $\int_{\gamma}f(z)dz:=\int_{a}^{b}f(z(t))z'(t)dt$.

I definitely prefer the first approach, because I think it gives a better understanding of what complex integration really is. However, most books take the second approach. Could you recommend a complex analysis book that takes the first approach?

An example of the first approach (the author first defined $\int_{\gamma}f(z)dz:=\int_{a}^{b}f(z(t))z'(t)dt$. But then he also described the first approach as follows.)
enter image description here

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    $\begingroup$ I'm failing to see the distinction you are making. In your second case, $f$ and $z'$ are complex valued, and you are calculating a complex valued Riemann integral. In your first case, meanwhile, you need to integrate a function along a curve; what other interpretation do have for that, other than the second meaning? $\endgroup$ Commented Apr 8 at 4:25
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    $\begingroup$ What book is your example from? $\endgroup$ Commented Apr 8 at 5:06
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    $\begingroup$ @littleO This example is from "Complex Analysis" by Kunihiko Kodaira. $\endgroup$ Commented Apr 8 at 5:07
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    $\begingroup$ @tchappyha: I still don't follow. What you posted is a deduction that a reasonable definition of integration along a curve is the usual definition of integral along a curve $$\int_\gamma f(z)\,dz=\int_a^b f(\gamma(t))\,\gamma'(t)\,dt.$$ I wouldn't be surprised if some texts don't include the deduction, as it could be expected that the formula is familiar from calculus. But it's hardly a "different approach". $\endgroup$ Commented Apr 8 at 6:55
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    $\begingroup$ Once you know about line integrals and differential forms, it’s not so helpful to go back to Riemann sums to define a complex line integral. It’s going backwards, not forwards. $\endgroup$ Commented Apr 8 at 17:59

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