7E (Further Complex Methods)

With the stated sign convention, write

t = y - x

to obtain

H (e^{iω x}) (y) = \frac{e ^{iω y}}{π} P \int_{- \infty}^{\infty} \frac{e ^{- iω t}}{t} d t .

P \int_{- \infty}^{\infty} \frac{e ^{- iω t}}{t} d t = - iπ sgn (ω) .

- i sgn (ω)

. Taking the real part yields

H (cos ω x) (y) = sgn (ω) sin (ω y) .

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