7E (Further Complex Methods)

There is an important convention issue in the printed statement. The ordinary Cauchy principal value exists only for

n = 1

. Closing the Fourier contour in the lower half-plane, with a small indentation above the real-axis pole, gives

PV \int_{R} \frac{e ^{- i x}}{x} d x = - iπ .

Equivalently, the odd cosine contribution cancels and the Dirichlet integral

\int_{0}^{\infty} sin x / x d x = π /2

supplies the imaginary part.

For

n \geq 2

, the paired integrand near zero is

2 cos x / x^{n}

n

is even and

- 2 i sin x / x^{n}

n

is odd. Its leading even power is nonintegrable:

x^{- n}

in the first case and

x^{1 - n}

in the second. Thus the ordinary principal value diverges for every $n > 1$ .

If the intended convention is the Hadamard finite-part integral, define it consistently by distributional differentiation,

Pf \frac{1}{x ^{n}} = \frac{( - 1 ) ^{n - 1}}{( n - 1 )!} \frac{d ^{n - 1}}{d x ^{n - 1}} PV \frac{1}{x} .

Integration by parts in this distributional pairing, or regularized Fourier transformation, then gives

Pf \int_{R} \frac{e ^{- i x}}{x ^{n}} d x = \frac{π ( - i ) ^{n}}{( n - 1 )!} .

The distinction is essential: subtracting the divergent local Taylor terms supplies a finite part, but symmetric exclusion alone does not.

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