Principal value integral I am reading A. Zee, QFT in a nutshell, and in appendix 1 he has:

Meanwhile the principal value integral is defined by:
  $$\int dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int dx\, {x\over x^2+\epsilon^2}f(x)$$

Please can someone explain to me why this is the case? As I understood it the principal value integral is rather defined as
$$\int_a^b dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0^+} \int_a^{-\epsilon} dx\, {1\over x}f(x)+\lim_{\epsilon \rightarrow 0^+} \int_{\epsilon}^b dx\, {1\over x}f(x),$$
where $a<0<b$. But as far as I can see these two definitions are not equivalent.
 A: Note that the right spelling is "principal value".
The formulae aren't identical but the results are the same whenever both definitions yield a well-defined expression. What matters is that we remove the leading logarithmic divergence on both sides from $x=0$ and we do so in a symmetric way with respect to $x\to -x$.
If you denote the second definition-based integral $Cut(\epsilon)$,
$$ Cut(\epsilon) = \left(\int_{a}^{-\epsilon}+\int_{\epsilon}^b\right) \frac{dx}x f(x)$$
then I claim that there exists a weighting function $g(y)$ such that
$$\int_0^{\infty} g(y) Cut(y) dy = \int dx\,\frac{x}{x^2+\epsilon^2}f(x) $$
so it reduces to the first definition-based integral. The function $g(y)$ is supported for $y$ of the same order as $\epsilon$ so the limit has the same effect on both expressions.
You may also see the equivalence of both expressions if you just Taylor-expand $f(x)$ near zero. Assuming that $f(x)$ is finite and well-behaved near $x=0$, it's easy to prove that both definitions yield the same result. The real purpose of the "principal value" terminology deals with branches of functions of complex variables. So you may also imagine that $f(x)$ is a meromorphic or holomorphic function of a complex $x$. The indefinite integral $\int f(x)/x$ has a logarithmic singularity around $x=0$ and one needs to define on which branch we are at. The principal value takes the average of the results one would get on the $+i\pi$ and $-i\pi$ branches for the logarithm of the negative numbers.
A: The short answer is that the two principal value definitions agree on sufficiently well-behaved functions, but may disagree on sufficiently singular functions. For instance,
on one hand
$$\lim_{\epsilon\searrow 0} \int_{\mathbb{R}\backslash[-\epsilon,\epsilon]} \frac{\mathrm{d}x}{x^3}~=~0$$
is zero, while on the other hand 
$$\lim_{\epsilon\searrow 0} \int_{\mathbb{R}} \frac{\mathrm{d}x}{x(x^2+\epsilon^2)}$$
is not well-defined, since the integrand is not integrable at $x=0$.
I) Here we would like to investigate further the definition of principal value $P\int\! \mathrm{d} x$. 

Definition. Let $\chi=(\chi_{\epsilon})_{\epsilon>0}$ be a family of functions $\chi_{\epsilon}:\mathbb{R}\to [0,1]\subseteq \mathbb{R}$ that are:
  
  
*
  
*even functions $\chi_{\epsilon}(x)~=~\chi_{\epsilon}(-x),$
  
*Lebesgue measurable functions, 
  
*$\chi_{\epsilon}(x)\nearrow 1$ pointwise almost everywhere for $\epsilon \searrow 0$.
  
  
  Let us refer to such a function $\chi_{\epsilon}$ as a kernel function.

Examples of kernel functions $\chi_{\epsilon}$ are for instance:


*

*the characteristic function 
$$\chi_{\epsilon}(x) ~=~\chi^{\rm std}_{\epsilon}(x) ~:=~ 1_{\mathbb{R}\backslash[-\epsilon,\epsilon]}(x)$$ 
for the set $\mathbb{R}\backslash[-\epsilon,\epsilon]$. (This choice $\chi^{\rm std}$ will lead to the standard definition of principal value.)

*the continuous function 
$$\chi_{\epsilon}(x) ~=~ \chi^{a,b}_{\epsilon}(x) ~:=~ \frac{|x|^a}{|x|^a+ \epsilon^b},$$ 
where $a,b>0$ are two positive constants. (The choice $\chi^{2,2}$ will lead to the other definition of principal value mentioned by OP.) 

*the constant unit function $\chi_{\epsilon}(x) ~=~ 1$. (Unsurprisingly, this latter choice will turn out to be not so useful.)
II) 

Definition. Define the set $V(\chi)$ of $\chi$-admissible functions as
  $$V(\chi)~:=~\left\{ f: \mathbb{R} \to  \mathbb{C} ~\left|~ \begin{array}{c}
f~\text{is Lebesgue measurable},\cr
\forall \epsilon>0~:~~ \chi_{\epsilon} f~\in~ {\cal L}^1(\mathbb{R}), 
\cr \text{and} \cr 
\left(\int\! \mathrm{d}x~ \chi_{\epsilon}(x) f(x)\right)_{\epsilon>0}  \text{is convergent for}~ \epsilon\searrow 0 \end{array}\right.\right\}. $$ 
Definition. If a function $f\in V(\chi)$ is $\chi$-admissible, we define the $\chi$-based principal value as
  $$P(\chi)\int\! \mathrm{d} x f(x)~:=~\lim_{\epsilon\searrow 0} \int\! \mathrm{d}x~ \chi_{\epsilon}(x) f(x).$$

Here ${\cal L}^1(\mathbb{R})$ denotes the set of functions that are Lebesgue integrable, i.e. functions that are Lebesgue measurable and whose absolute value has a finite integral. ${\cal L}^1(\mathbb{R})$ is an example of an ${\cal L}^p$ space.
III) It is not hard to see that: 


*

*If $f\in{\cal L}^1(\mathbb{R})$ is Lebesgue integrable, then it is $\chi$-admissible $f\in V(\chi)$, and the principal value
$$P(\chi)\int\! \mathrm{d} x ~f(x)~=~ \int\! \mathrm{d} x ~f(x)$$ 
is just the ordinary Lebesgue integral because of the 
Lebesgue dominated convergence theorem.

*The set $V(\chi)$ of $\chi$-admissible functions is a $\mathbb{C}$-vector space.

*If a function $f\in V(\chi)$ is $\chi$-admissible, so is the mirrored function $(x\mapsto f(-x))\in V(\chi)$, with same principal value.

*If an $\chi$-admissible function $f\in V(\chi)$ is odd, then $P(\chi)\int\! \mathrm{d} x~f(x) ~=~ 0$. 
Thus it is enough to investigate even and odd functions.
Finally, let us investigate power functions $x\mapsto x^p$, $p\in\mathbb{R}$, which play an important role in practice as building blocks.
IV) Even functions. Let 
$$g_{p,K}(x) ~:=~ 1_{[-K,K]}(x) |x|^p~=~g_{p,K}(-x)$$ 
be a truncated power function, where $p\in\mathbb{R}$ is a real power, and where $K>0$ is a positive truncation constant. 
It is not hard to show that in the case of Example 1, 2, or 3,
$$g_{p,K}\in V(\chi) \qquad \Leftrightarrow \qquad p>-1\qquad \Leftrightarrow \qquad g_{p,K}\in {\cal L}^1(\mathbb{R}).$$
In the affirmative case $p>-1$, the principal value definitions based on the three Examples 1, 2, and 3 agree: 
$$P(\chi)\int\! \mathrm{d} x ~g_{p,K}~=~\int\! \mathrm{d} x ~g_{p,K}~=~ \frac{2K^{p+1}}{p+1}.$$
V) Odd functions. Let 
$$h_{p,K}(x) ~:=~ {\rm sgn}(x) 1_{[-K,K]}(x) |x|^p~=~-h_{p,K}(-x)$$ 
be a truncated power function, where $p\in\mathbb{R}$ is a real power, and where $K>0$ is a positive truncation constant. In the three Examples 1, 2, and 3, we get


*

*$h_{p,K}\in V(\chi^{\rm std})$ always,

*$h_{p,K}\in V(\chi^{a,b}) \qquad \Leftrightarrow \qquad p+a>-1$,

*$h_{p,K}\in V(1) \qquad \Leftrightarrow \qquad p>-1\qquad \Leftrightarrow \qquad h_{p,K}\in {\cal L}^1(\mathbb{R}).$

A: There are two issues here which recur frequently on this forum--- which functions can be regarded as distributions and what is the nature of convergence in the sense of distributions?  Perhaps a mathematician's take might be of interest.  It is standard that locally integrable functions can be regarded as distributions in a natural way, less so that the same applies to  meromorphic functions (in the OP $\frac 1x$).  The method is the same in both cases but is particularly simple in the one in hand.   The function $\ln  |x|$ is locally integrable and so a distribution.  It has a distributional derivative and it is natural to define the distribution $\frac 1x$ to be this derivative.  One can show quite easily that it then has all of the properties that one expects (including those of the approach using principal parts).  
As to convergence, I will forego the precise definition.  For almost all practical purposes, it suffices to know 2 facts. Firstly, if a  sequence of functions converges locally in the $L^1$-norm, then it converges in the sense of distributions and, secondly, we can always differentiate such a convergent family and convergence is retained (the dream theorem of every calculus student).
After this preparation, the question in hand has a one line proof.  The family $\frac 12 \ln (x^2+\epsilon^2)$ converges to $\ln|x|$ in the above sense and differentiating provides the required formula.
All the facts about distributions used here can be found in the monograph of L. Schwartz but I can recommend the elementary approach of J. Sebastiao e Silva which is expounded in "An Introduction to the Theory of Distributions" by Campos Ferreira. This uses only the tools of elementary one-dimensional calculus.
