Intereting Posts

How are Zeta function values calculated from within the Critical Strip?
A question related to the card game “Set”
Variation of the Kempner series – convergence of series $\sum\frac{1}{n}$ where $9$ is not a digit of $1/n$.
On order of elements of a infinite group
Show that a graph has a unique MST if all edges have distinct weights
A question on coalgebras(1)
Why is the volume of a cone one third of the volume of a cylinder?
Integral basis of an extension of number fields
Small Representations of $2016$
Locally Constant Functions on Connected Spaces are Constant
set theory proof of $A\cap B = A \setminus(A\setminus B)$
If $f^2$ is an analytic function then so is $f$
For which primes p is $p^2 + 2$ also prime?
Characteristic of an integral domain must be either $0$ or a prime number.
Integrate $\frac{x^2-1}{x^2+1}\frac{1}{\sqrt{1+x^4}}dx$

My calculus book says that the integral of $\frac1x$ cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and opposite insign, so cancel, even though they do not converge.

Now typing into wolframalpha I got “does not converge” and only at the bottom did the expected result appeared ($\ln|b|-\ln|a|$) as “Cauchy principal value” (CPV, which I looked up on wikipedia). Why that fancyness? Does this has some implications in some application area?

And also, when I ask wolframalpha about “integral of cos/sin from $0$ to $\frac\pi2$”, I get infinity (intuitive, looking at the plot, although $\ln|sin(\frac\pi2)|-\ln|sin(0)|$ is admittedly wrong ) as “CPV”, but when I ask for “integral of cos/sin from $0$ to $\pi$” which I expect to be zero, because the plot is symmetric/odd I get “does not converge” and there is no CPV result either. Why?

- Equicontinuity of $x^n$
- Factorial and exponential dual identities
- Solutions to $f'=f$ over the rationals
- Family of Closed/Open Nested Intervals
- Big Rudin 1.40: Open Set is a countable union of closed disks?
- Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $$

- Question about Feller's book on the Central Limit Theorem
- Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$
- Pointwise but not uniform convergence of a Fourier series
- A twice continuously differentiable function
- Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$
- Why sum of two little “o” notation is equal little “o” notation from sum?
- Some integral with sine
- Is it true that a space-filling curve cannot be injective everywhere?
- Convergence of $\sum_{k=1}^{n} f(k) - \int_{1}^{n} f(x) dx$
- Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$

Regarding your original question: let $f(x)$ be an odd function on $\mathbb R$. We can think of two different ways of computing the improper integral

$$\int_{-\infty}^\infty f(x) dx=\lim_{a\to\infty}\int_{-a}^a f(x) dx,$$

or

$$\int_{-\infty}^\infty f(x) dx=\lim_{a\to -\infty, b\to\infty}\int_a^b f(x) dx.$$

Since $f$ is odd, the first way gives $0$; this is what is called Cauchy Principal Value. However, the proper way to compute the improper integral is the second way. The reason why, is because we want (in some sense) to take the supremum over all intervals $[a,b]\subset \mathbb R$. And this way of computing it is the reason why $f(x)=\dfrac{1}{x}$ (or $f(x)=x$) will give divergent integrals.

The Cauchy principal value and other pseudo-functions are common in distribution theory (not only…) and in physics where infinite values are removed nicely when possible (hidden under the carpet with renormalization tricks else… ;-)).

In distribution theory we may always exchange derivation and limit, all derivatives remain distributions, every distribution admits one primitive (up to a *global* constant) and so on… For these (generous) rules to hold we need tolerant functions so that the derivative of $\log|x|$ for example will be P.V. $\frac 1x$ and not simply $\frac 1x$ (the integral of the last one would not exist !). Curiously multiplication became more difficult when multiplying two $\delta(x)$ for example (even with Colombeau and others ideas) but we can’t have everything…

Distribution theory is much used in advanced part of physics (with great names like Sobolev, Gel’fand, Dirac and later Schwartz bringing the mathematical respectability) because it allows for example to handle discrete values as well as continuous spectra in an unified way but let’s stop the propaganda and come to your second part…

Concerning your “integral of cos/sin from 0 to pi” not evaluated by Alpha. Well it seems that the software didn’t considerer compensation of singularities at two finite points (I don’t know if Alpha handles compensations at $-\infty$ and $+\infty$ but it could consider them as the same ‘point’ $\infty=\frac 10$). For physical application you would have to specify the limits say ‘from $\epsilon$ to $\pi-\epsilon$’ as $\epsilon \to 0$.

Regarding the problem $$\int_a^b{\frac{1}{x}dx}$$ the fundamental theorem of calculus requires that the function be continuous over the interval $(a, b)$, and while $f(x) = \frac{1}{x}$ is continuous *over its domain* (which is $(-\infty, 0)\cup (0, +\infty)$), it is not continuous over an interval that crosses 0. So the integral can be found using CPV.

As for integrating $\cot(x)$ over the interval $(0, \frac{\pi}{2})$, note that $\sin{(0)} = 0$, and the domain of logarithmic functions is strictly positive. Equivalently,

$$\ln{\left|\sin{\left(\frac{\pi}{2}\right)}\right|} – \ln{|\sin(0)|} = \ln{\left|\frac{\sin{\left(\frac{\pi}{2}\right)}}{\sin{(0})}\right|}$$

which causes division by zero, so can’t be used.

- Commutator of Vector Fields
- $\mathbb{A}^{2}$ not isomorphic to affine space minus the origin
- Endomorphism rings and torsion subgroups.
- On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$
- 3D coordinates of circle center given three point on the circle.
- Why don't we define “imaginary” numbers for every “impossibility”?
- closed bounded subset in metric space not compact
- Classifying the compact subsets of $L^p$
- What is the order when doing $x^{y^z}$ and why?
- If a rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$
- What are the rings in which left and right zero divisors coincide called?
- How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?
- Finding the Moment Generating function of a Binomial Distribution
- Prove $\forall a,b,c \in \mathbb Z$, if $ab+ac\equiv 3\pmod 6$ then $b \not\equiv c \pmod 6$ by contraposition
- How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)