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An interior Cauchy principal value reduces to a regular integral

Statement

For $f (x) = x$ on $[0, 1]$ and $c = 1 / 3$ , the symmetric Cauchy principal value is $1 + \frac{1}{3} \log 2$ .

Facts & Assumptions

Given: $f (x) = x$ , $a = 0$ , $c = 1 / 3$ , $b = 1$ .

[F1]

The Cauchy principal-value subtraction identity: for $f \in C^{1} ([a, b])$ , the principal value is the ordinary integral of $(f (x) - f (c)) / (x - c)$ plus $f (c) \log ((b - c) / (c - a))$ .

[F2]

The natural logarithm as the inverse of the exponential function and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is the natural logarithm and $\log 2$ is defined for the positive number $2$ .

[F3]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : the linear function is $C^{1}$ on $[0, 1]$ .

Proof

1.1givenF3algebra

The function $f (x) = x$ is $C^{1}$ by [F3] and $f (c) = 1 / 3$ . For every $x \neq c$ , $(f (x) - f (c)) / (x - c) = 1$ , and its continuous extension at $c$ is also $1$ .

2.1F1F2step 1.1algebra∎

Apply [F1]. The ordinary integral of this quotient is $\int_{0}^{1} 1 d x = 1$ , and $(b - c) / (c - a) = (2 / 3) / (1 / 3) = 2$ . Thus the principal value equals $1 + (1 / 3) \log 2$ .

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

Lloyd N. Trefethen and J. A. C. Weideman, The Exponentially Convergent Trapezoidal Rule, SIAM Review 56(3), 2014 (standard reference, not scraped)