Hadamard regularization

Template:Renormalization and 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 Template:Harvs. Template:Harvs showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.

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

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

The Hadamard finite part integral above (for Template:Math) may also be given by the following equivalent definitions: $ℋ \int_{a}^{b} \frac{f (t)}{(t - x)^{2}} d t = \lim_{ε \to 0^{+}} {\int_{a}^{x - ε} \frac{f (t)}{(t - x)^{2}} d t + \int_{x + ε}^{b} \frac{f (t)}{(t - x)^{2}} d t - \frac{f (x + ε) + f (x - ε)}{ε}},$ $ℋ \int_{a}^{b} \frac{f (t)}{(t - x)^{2}} d t = \lim_{ε \to 0^{+}} {\int_{a}^{b} \frac{(t - x)^{2} f (t)}{((t - x)^{2} + ε^{2})^{2}} d t - \frac{π f (x)}{2 ε} - \frac{f (x)}{2} (\frac{1}{b - x} - \frac{1}{a - x})} .$

The definitions above may be derived by assuming that the function Template:Math is differentiable infinitely many times at Template:Math, that is, by assuming that Template:Math can be represented by its Taylor series about Template:Math. For details, see Template:Harvs. (Note that the term Template:Math in the second equivalent definition above is missing in Template:Harvs but this is corrected in the errata sheet of the book.)

Integral equations containing Hadamard finite part integrals (with Template:Math 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 $\int_{- 1}^{1} \frac{1}{t^{2}} d t = (\lim_{a \to 0^{-}} \int_{- 1}^{a} \frac{1}{t^{2}} d t) + (\lim_{b \to 0^{+}} \int_{b}^{1} \frac{1}{t^{2}} d t) = \lim_{a \to 0^{-}} (- \frac{1}{a} - 1) + \lim_{b \to 0^{+}} (- 1 + \frac{1}{b}) = + \infty$ Its Cauchy principal value also diverges since $𝒞 \int_{- 1}^{1} \frac{1}{t^{2}} d t = \lim_{ε \to 0^{+}} (\int_{- 1}^{- ε} \frac{1}{t^{2}} d t + \int_{ε}^{1} \frac{1}{t^{2}} d t) = \lim_{ε \to 0^{+}} (\frac{1}{ε} - 1 - 1 + \frac{1}{ε}) = + \infty$ To assign a finite value to this divergent integral, we may consider $ℋ \int_{- 1}^{1} \frac{1}{t^{2}} d t = ℋ \int_{- 1}^{1} \frac{1}{(t - x)^{2}} d t |_{x = 0} = \frac{d}{d x} (𝒞 \int_{- 1}^{1} \frac{1}{t - x} d t) |_{x = 0}$ The inner Cauchy principal value is given by $𝒞 \int_{- 1}^{1} \frac{1}{t - x} d t = \lim_{ε \to 0^{+}} (\int_{- 1}^{- ε} \frac{1}{t - x} d t + \int_{ε}^{1} \frac{1}{t - x} d t) = \lim_{ε \to 0^{+}} (\ln | \frac{ε + x}{1 + x} | + \ln | \frac{1 - x}{ε - x} |) = \ln | \frac{1 - x}{1 + x} |$ Therefore, $ℋ \int_{- 1}^{1} \frac{1}{t^{2}} d t = \frac{d}{d x} (\ln | \frac{1 - x}{1 + x} |) |_{x = 0} = \frac{2}{x^{2} - 1} |_{x = 0} = - 2$ Note that this value does not represent the area under the curve Template:Math, which is clearly always positive. However, it can be seen where this comes from. Recall the Cauchy principal value of this integral, when evaluated at the endpoints, took the form $\lim_{ε \to 0^{+}} (\frac{1}{ε} - 1 - 1 + \frac{1}{ε}) = + \infty$

If one removes the infinite components, the pair of $\frac{1}{ε}$ terms, that which remains is $\lim_{ε \to 0^{+}} (- 1 - 1) = - 2$

which equals the value derived above.

References

Hadamard regularization

Example

References

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