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In Complex Analysis texts, one often sees integrals of functions $f : A \subset \mathbb{C} \to \mathbb{C}$ along contours $\gamma : [0,1] \to \mathbb{C}$.

But I've never seen discussion of integrals of functions $f : A \subset \mathbb{C} \to \mathbb{C}$ over domains $D \subset A$ in any of these texts.

It seems like it should be possible to define such integrals, so why aren't they discussed in most complex analysis textbooks? Are there any major results about them like, for example, Cauchy's integral formula for contour integrals? Are there any textbooks that do delve into these kinds of integrals?

Examples

By a contour integral, I mean something like the following: consider the contour $\gamma : [0,1] \to \mathbb{C}$ given by $\gamma(t) = e^{2\pi i t}$ and the function $f : \mathbb{C} - \{0\} \to \mathbb{C}$ given by $f(z) = \frac{1}{z}$. The integral of $f$ around the contour $\gamma$ is $$ \int_\gamma f = \int_0^1 f(\gamma(t))\gamma'(t) dt = \int_0^1 \frac{1}{e^{2\pi i t}} \frac{2\pi i e^{2\pi i t}}{1} dt = \int_0^1 2\pi i dt = 2\pi i $$ By an integral over a domain, I mean something like the following: consider the function $g : \mathbb{C} \to \mathbb{C}$ given by $g(z) = z^2$ and the domain $D = \{z \in \mathbb{C} : |z| \leq 1\}$. The integral of $g$ over the domain $D$ is $$ \int_D g d\mu \stackrel{?}{=} $$ where, presumably, $d\mu$ is something like the Haar measure on $\mathbb{C}$ with $\mu([0,1]\times[0,1]) = 1$.

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  • $\begingroup$ I don't see the difference between the first and the second category. Maybe an example would help ? $\endgroup$ – Jean Marie Aug 22 at 13:49
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    $\begingroup$ I'll add one to the question. $\endgroup$ – Charles Hudgins Aug 22 at 14:23
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The reason contour integrals are useful in complex analysis is due to Cauchy theorem and all its corollaries like the computation of integrals with residues, the Cauchy bounds for derivatives of all orders, maximum modulus theorem, very strong results about locally uniform convergence of holomorphic functions (such can be differentiated term by term for example which is not the case for uniform convergence of even real analytic functions) etc.

2-dimensional integrals have their use here and there in Complex Analysis (see Area Theorem for univalent functions which leads to the original Bieberbach estimate of the second coefficient) and various other specialized topics like the Bergman kernel and various spaces related to the Hardy spaces but defined by area integrals bounds, however, they are considerably more important in Real Analysis topics (in 2 variables here)

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  • $\begingroup$ You write: "2-dimensional integrals have their use here and there in Complex Analysis" OP asks why they aren't treated in Complex Analysis texts. $\endgroup$ – nilo de roock Aug 22 at 17:01
  • $\begingroup$ Explained why - they appear in specialized topics that are usually not treated in standard texts that do the basics (Rudin,Ahlfors, Conway etc) , but for example if you want to study univalent functions (also called geometric function theory) or if you want to study inner functions for example you will encounter them in important roles. Otherwise they are part of the real theory since there is nothing complex intrinsic to the area element as it contains both $dz, d\bar z$, so you cannot separate the holomorphic part like you can do in line integrals. $\endgroup$ – Conrad Aug 22 at 18:33
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    $\begingroup$ In short, area integrals are not important for the basic theory of complex functions. $\endgroup$ – Conrad Aug 22 at 18:39

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