complex analysis - Higher dimensional analogues of the argument principle?

I know there are higher dimensional analogues of the argument principle.

(See http://en.wikipedia.org/wiki/Variation_of_argument)

But I do not have books about it and I cannot find anything of value on the internet for free.

Plz give me references or explainations.

I would like an example of how one can find the zero's of functions in 3 Dimensions with an integral.

I believe it works as follows :

1) The function needs to be "3D-analytic" , and by that I mean the derivative can be given by the limit

$$ \lim_{h->0} \frac{f(x+h)-f(x)}{h} $$

Where any infinitesimal $h$ will give the same limit as any other infinitesimal $h_2$.
(By "any" I mean in any direction of the dimension.)

This is the natural analogue of complex differentiable.

2) The integral is taken over a closed surface that contains a volume.
The surface must be topologically isomorphic to a sphere. ( no holes , no selfintersections , continu )

3) Just like the argument principle , the direction of the integral matters.
Hence we riemann sum the points with respect to the direction we are going in.
( compare : for contour integration if our (piecewise) path goes from - to + we add if it goes from + to - we substract )

4) The "3D argument principle" then computes $CZ$.
$C$ is some constant and $Z$ is the number of zero's - the number of poles within the volume.

Is this correct ?

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The analytic proof of the PNT constructs a counting function for the primes based on the zero's of the Riemann zeta function.

An important part of that proof is the argument principle.

I wonder if there are some similar applications of the "3D argument principle " as described above that have been used in number theory.

I wonder what functions in 3D are associated with (number-theoretical) counting functions*.

(* such as used in the proof of PNT with the argument principle (used on the complex plane).)

Answer

Of course, the Argument Principle is in the context of meromorphic functions. However, the higher-dimensional analogue in the context of smooth functions comes from the theory of the degree of maps and winding numbers.

In particular, suppose $W$ is a compact $n$-dimensional oriented manifold with boundary, $f\colon W \to \Bbb R^n$ is smooth, and $f\big|_{\partial W} \ne 0$. If $0$ is a regular value of $f$, then $f^{-1}(0)$ consists of a finite number of points $x_1,\dots,x_k\in W$, each of which appears with a sign $\epsilon_j$ ($+1$ if $df_{x_j}\colon T_{x_j}W\to\Bbb R^n$ is orientation-preserving, and $-1$ if it is orientation-reversing). Then
$$\sum_{j=1}^k \epsilon_j = \text{deg}\left(\frac{f}{\|f\|}\Bigg|_{\partial W}\colon \partial W\to S^{n-1}\right)\,.$$

Comment: In the case $n=2$, $W\subset\Bbb R^2$, and $f$ holomorphic, the $\epsilon_j$ are all always $+1$, and so we merely count the roots in $W$. The degree on the right-hand side is the winding number of $f(\partial W)$, which is exactly what $\dfrac1{2\pi i}\displaystyle\int_{\partial W} \frac{f'(z)}{f(z)}dz$ computes.

The best reference I know on such matters is Guillemin and Pollack's Differential Topology. See pp. 110-111 and 144, in particular.

EDIT: To respond to your additional comments/questions, the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. Once you move on to $\Bbb R^n$, the derivative becomes a linear transformation with a limit property but cannot itself be given by the sort of limit you desire. Only directional derivatives have the single-variable calculus definition. (In time, you will learn some rigorously done multivariable calculus.) The degree calculation I have given above can indeed be represented by an integral (but, once again, you'll have a bit to learn to understand this): Choose an $(n-1)$-form $\omega$ on $S^{n-1}$ with $\displaystyle\int_{S^{n-1}}\omega=1$, and compute $\displaystyle\int_{\partial W} \left(\tfrac{f}{\|f\|}\right)^*\omega$. Alternatively, compute
$$\tfrac1{\text{vol}(S^{n-1})}\int_{\partial W} f^*\left(\tfrac{x_1\, dx_2\wedge\dots\wedge dx_n - x_2\, dx_1\wedge dx_3\wedge \dots \wedge dx_n + (-1)^{n-1} x_n\, dx_1\wedge\dots\wedge dx_{n-1}}{(x_1^2+x_2^2+\dots+x_n^2)^{n/2}}\right)\,.$$
This is in fact an immediate generalization of the integral appearing in the argument principle.

There are generalizations of the Residue Theorem (which is where the Argument Principle comes from) to meromorphic functions in $\Bbb C^n$. This is also a rather rich subject; some comments were made in this post.