Algebraic Geometry

Download e-book for iPad: Hilbert transforms: by Frederick W. King

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By Frederick W. King

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9 Z transform (one-sided or two-sided); Eqs. 8, HT-01 spherical Bessel function of the second kind; Eq. 116) z z¯ z∗ zI Z Z+ Z{xn } complex variable, z = x + iy ; Eq. 67) complex conjugate of z complex conjugate of z inverse point (or image point) of z; Eq. 35) set of integers 0, ±1, ±2, . . set of non-negative integers 0, 1, 2, . . 6, Eq. 14 Zs (ω) surface impedance function; Eq. 1, Eq. 6) α(ω) absorption coefficient of a medium; Eq. 915 965 594 177 219 015 1 . 577 215 664 9 . ) contour in the complex plane (frequently used to signify a non-closed contour) (z) gamma function; Eq.

Such equations are referred to as singular integral equations. The Hilbert transform on R, the real line, is defined by Hf (x) = 1 P π ∞ −∞ f ( y)dy , x−y for x ∈ R. 3) which is singular when y = x. The symbol P denotes an extension of the normal definition of the integral called the Cauchy principal value. This is discussed in detail in Chapter 2. The integral becomes well behaved for many common functions if an infinitesimally small section of the integration interval centered at the singularity y = x is deleted, as part of the definition of the integral.

Occasionally the Hilbert transform is defined with the factor π −1 omitted. Employing the definition given in Eq. 2) does have the advantage that factors of π that would frequently appear are incorporated into the definition of the Hilbert transform. A few authors define the Hilbert transform with the imaginary unit factor included, that is, π −1 is replaced by (π i)−1 . Note that nothing has been said about what conditions must be specified for the function f in order that the integral in Eq. 2) exists.

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