Hadamard regularization

In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by. showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.

If the Cauchy principal value integral $$\mathcal{C}\int_a^b \frac{f(t)}{t-x} \, dt \quad (\text{for } a<x<b)$$ exists, then it may be differentiated with respect to $x$ to obtain the Hadamard finite part integral as follows: $$\frac{d}{dx} \left(\mathcal{C}\int_{a}^{b} \frac{f(t)}{t-x} \,dt\right)=\mathcal{H}\int_a^b \frac{f(t)}{(t-x)^2}\, dt \quad (\text{for } a<x<b).$$

Note that the symbols $$\mathcal{C}$$ and $$\mathcal{H}$$ are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively.

The Hadamard finite part integral above (for $a &lt; x &lt; b$) may also be given by the following equivalent definitions: $$\mathcal{H}\int_a^b \frac{f(t)}{(t-x)^2}\, dt = \lim_{\varepsilon \to 0^+} \left\{ \int_a^{x-\varepsilon} \frac{f(t)}{(t-x)^2} \, dt + \int_{x+\varepsilon}^b\frac{f(t)}{(t-x)^2}\,dt -\frac{f(x+\varepsilon)+f(x-\varepsilon)}{\varepsilon}\right\},$$ $$\mathcal{H}\int_a^b \frac{f(t)}{(t-x)^2}\, dt = \lim_{\varepsilon \to 0^+} \left\{ \int_a^b \frac{(t-x)^{2}f(t)} {((t-x)^2+\varepsilon^2)^2}\,dt -\frac{\pi f(x)}{2\varepsilon}- \frac{f(x)}{2} \left(\frac{1}{b-x}-\frac{1}{a-x}\right)\right\}. $$

The definitions above may be derived by assuming that the function $f&thinsp;(t)$ is differentiable infinitely many times at $t = x for a &lt; x &lt; b$, that is, by assuming that $f&thinsp;(t)$ can be represented by its Taylor series about $t = x$. For details, see. (Note that the term $− f&thinsp;(x)⁄2(1⁄b − x − 1⁄a − x)$ in the second equivalent definition above is missing in but this is corrected in the errata sheet of the book.)

Integral equations containing Hadamard finite part integrals (with $f&thinsp;(t)$ unknown) are termed hypersingular integral equations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as in fracture analysis.

Example
Consider the divergent integral $$\int_{-1}^1 \frac{1}{t^2} \, dt = \left( \lim_{a \to 0^-} \int_{-1}^{a} \frac{1}{t^2} \, dt \right) + \left( \lim_{b \to 0^+} \int_{b}^{1} \frac{1}{t^2} \, dt \right) = \lim_{a\to 0^-} \left( -\frac{1}{a} - 1 \right) + \lim_{b\to 0^+} \left(-1 + \frac{1}{b}\right)= +\infty$$ Its Cauchy principal value also diverges since $$\mathcal{C} \int_{-1}^1 \frac{1}{t^2} \, dt = \lim_{\varepsilon \to 0^+} \left( \int_{-1}^{-\varepsilon} \frac{1}{t^2} \, dt + \int_{\varepsilon}^1 \frac{1}{t^2} \, dt \right) = \lim_{\varepsilon \to 0^+} \left( \frac{1}{\varepsilon} - 1 - 1 + \frac{1}{\varepsilon} \right) = +\infty $$ To assign a finite value to this divergent integral, we may consider $$\mathcal{H}\int_{-1}^1 \frac{1}{t^2} \, dt = \mathcal{H} \int_{-1}^1 \frac{1}{(t-x)^2} \, dt \Bigg|_{x=0} = \frac{d}{dx}\left( \mathcal{C} \int_{-1}^1 \frac{1}{t-x} \, dt \right) \Bigg|_{x=0}$$ The inner Cauchy principal value is given by $$\mathcal{C} \int_{-1}^1 \frac{1}{t-x} \, dt = \lim_{\varepsilon \to 0^+} \left( \int_{-1}^{-\varepsilon} \frac{1}{t-x} \, dt + \int_{\varepsilon}^1 \frac{1}{t-x} \, dt \right) = \lim_{\varepsilon \to 0^+} \left( \ln\left| \frac{\varepsilon +x}{1+x} \right| + \ln \left| \frac{1-x}{\varepsilon - x} \right| \right) = \ln\left| \frac{1-x}{1+x} \right|$$ Therefore, $$\mathcal{H}\int_{-1}^1 \frac{1}{t^2} \, dt = \frac{d}{dx}\left( \ln\left| \frac{1-x}{1+x} \right| \right) \Bigg|_{x=0} = \frac{2}{x^2-1}\Bigg|_{x=0} = -2$$ Note that this value does not represent the area under the curve $y(t) = 1/t^{2}$, which is clearly always positive.