I decided to consider divergent (to infinity) integrals as some new kind of number. Towards this end, I began by establishing certain rules defining equivalence of the integrals. It seems the usual change of variable theorems do not generally work with divergent integrals, but (in addition to some usual linearity rules) the following rule involving Laplace transformations proved very useful:
$$\int_0^\infty f(x)dx=\int_0^\infty\mathcal{L}_t[t f(t)](x)dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)dx$$
In other words there is a transformation (and its inverse) which preserves the "area" under the integral.
I was able to derive various representations of the same infinite "numbers" using this transformation. Below is a list of some of the most notable of such numbers together with some of their properties. I would be glad to find out whether they appear in their various integral, series or other forms in other areas of mathematics or physics.
1. Symbol: $\tau$. Regularization (finite part):$0$. Determinant(Modulus):$\frac{e^{-\gamma}}4$
- Integral representations:
$$ \int_0^{\infty } \, dx=\int_0^{\infty } \frac{1}{x^2} \, dx$$
- Series representations:
$$\sum_{k=-\infty}^\infty \frac12=\sum_{k=-\infty}^{+\infty}\frac{(-1)^k+1}2=\sum_{k=-\infty}^{+\infty}\frac{(-1)^{k+1}+1}2$$
- Integral representations of powers ($n>0$):
$$\tau^n=B_n(1/2)+n\int_0^\infty B_{n-1}(x+1/2)dx$$ (Bernoulli polynomials).
- Other properties:
$$\operatorname{reg} \frac1\pi\ln \left(\frac{\tau +\frac{z}{\pi }}{\tau -\frac{z}{\pi }}\right)=\tan z$$
- Remarks: Formally due to Fourier transform of Dirac Delta function, can be represented as $\pi\delta(0)$. Constitutes half of the numerocity of integers, equals to the numerocities of even or odd numbers. May be considered as belonging to the (generalized) ring of periods, if that ring is allowed to include divergent integrals.
2. Symbol: $\omega_-, \tau-1/2$. Regularization (finite part): $-1/2$. Determinant(Modulus):not defined, principal value is $e^{-\gamma}$.
Integral representations: $$\int_{1/2}^\infty dx=\int_0^\infty \frac{e^{-\frac{x}{2}} (x+2)}{2 x^2}dx=\int_0^2\frac1{x^2} dx$$
Series representations: $$\sum_{k=1}^\infty 1 $$
Integral representation of powers: $$(\omega_-+a)^n=B_n(a)+n\int_0^\infty B_{n-1}(x+a)dx$$
Remarks: represents the numerocity of natural numbers. May be considered as belonging to the (generalized) ring of periods, if that ring is allowed to include divergent integrals.
3. Symbol: $\omega_+, \tau+1/2$. Regularization (finite part): $1/2$. Determinant(Modulus): $e^{-\gamma}$.
- Integral representations:
$$\int_{-1/2}^\infty dx=\int_0^\infty \frac{4-e^{-\frac{x}{2}} (x+2)}{2 x^2} dx$$
- Series representations:
$$\sum_{k=0}^\infty 1 $$
Other properties: $$\operatorname{reg}\ln \omega_+=\gamma$$
Remarks: represents the numerosity of non-negative integers. May be considered as belonging to the (generalized) ring of periods, if that ring is allowed to include divergent integrals.
There are properties, common to $\omega_+$ and $\omega_-$:
For $n>1$, $$\int_0^\infty x^n dx=\frac{\omega _+^{n+2}-\omega _-^{n+2}}{(n+1)(n+2)}$$
$$\int_0^\infty \frac1{x^n} dx=\frac{\omega _+^{n}-\omega _-^{n}}{(n-1)n!}$$
$$\operatorname{reg}\frac1{\pi }\ln \left(\frac{\omega _+-\frac{z}{\pi }}{\omega _-+\frac{z}{\pi }}\right)=\cot z$$
- And an expression for the derivative of an analytic function: $$f'(x)=\operatorname{reg}(f(\omega_++x)-f(\omega_-+x))=\operatorname{reg} \Delta f(\omega_-+x)$$
4. Symbol: $H, \ln \omega_++\gamma$. Regularization (finite part): $0$. Modulus: ?
- Integral representations:
$$\int_0^1 \frac1x dx-\gamma=\int_1^\infty \frac1xdx=\int_0^\infty\frac{e^{-x}}{x}dx=\int_0^\infty\frac{dx}{x+1}=\int_0^\infty\frac{e^x x \text{Ei}(-x)+1}{x}dx=\int_0^\infty\frac{x-\ln x-1}{(x-1)^2}dx$$
Series representation: $$\sum_{k=1}^\infty \frac1x-\gamma$$
Remarks: $H+\gamma/2$ is the area of hyperbolic sector of a unit hyperbola, corresponding to infinite hyperbolic angle.
5. Symbol: $H+\gamma, \ln\omega_++2\gamma$. Regularization (finite part): $\gamma$. Modulus: ?
- Integral representations:
$$\int_0^1 \frac1x dx=\int_0^\infty\frac{1-e^{-x}}{x}dx=\int_0^\infty\frac{1}{x^2+x}dx=-\int_0^\infty e^x \text{Ei}(-x)dx=-\int_0^\infty\frac{x-x\ln x-1}{(x-1)^2 x}dx$$
Series representation: $$\sum_{k=1}^\infty \frac1x$$
Remarks: May be considered as belonging to the (generalized) ring of periods, if that ring is allowed to include divergent integrals.
6. Symbol: $\frac {\tau^2}2-\frac1{24}-H-\frac\gamma 2$. Regularization (finite part): $-\gamma/2$
- Integral representation: $$\int_1^\infty \sqrt{x^2-1}dx=\int_0^\infty \frac{K_2(x)}{x}dx=\int_0^\infty \left(x-\frac{1}{2 x}\right) dx =\int_0^\infty \left(\frac{2}{x^3}-\frac{1}{2 x}\right)dx$$
7. Symbol: $e^{\omega_-}$. Regularization (finite part): $\frac1{e-1}$. Modulus: $\frac1{\sqrt{e}}$
- Integral representation: $$\frac1{e-1}+\frac1{e-1}\int_{-\infty}^\infty e^x dx$$
8. Symbol: $e^{\omega_+}$. Regularization (finite part): $\frac e{e-1}$. Modulus: $\sqrt{e}$
- Integral representation: $$\frac1{1-e^{-1}}+\frac1{1-e^{-1}}\int_{-\infty}^\infty e^x dx$$
9. Symbol: $(-1)^\tau$. Regularization (finite part): $\frac\pi2$. Modulus: $1$
- Remarks: No known divergent integral is known that corresponds to this expression. This may mean it is reasonable to introduce "divergent numbers" that generalize the notion of divergent integrals. We still can speak about the "finite part" and "modulus" here.
So, I gave a list of some constants that can be represented in the forms of different divergent integrals. I wonder, whether they appear in other fields. Notice that they show interesting relations with other mathematical constants, primarily the Euler-Mascheroni constant.