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4 Co-spectrum and quad-spectrum The real part α(ω) and the imaginary part b(co) of F(co), introduced by [ 1 6 ] , 41 BASIC THEOREMS FOR THE FOURIER T R A N S F O R M are also used individually and have special names: CO α(ώ) = J cos cot f(t) at — CO [28] CO ί>(ω) = j f(t) sin cot dt a(co) is the Fourier cosine transform of f(t) and is called co-spectrum. ό(ω) is the Fourier sine transform of / ( / ) and is called quadrature spectrum or quad-spectrum (cf. , 1959). , ib(co) is the sine transform of the function f(t) after a phase change of π/2 has been applied to each harmonic of f(t).

4. They are also useful in solving differential equations (cf. 223-227). 1 Transcription between different systems for Fourier transform and its inversion The calculation of the Fourier transform F(co) for any given function f(t) CALCULATION OF FOURIER TRANSFORMS OF ANALYTIC FUNCTIONS 49 has two different aspects: (1) f(t) is an analytically given function. In this case the calculation consists in purely mathematical operations, using [ 2 0 ] . (2) / ( / ) represents a series of observations. Case 2 is of the greatest significance in geophysical applications, and will be dealt with in following chapters.

The same deductions can easily be performed for fi — t), f*it), / * ( — 0 · F ° example, / * ( * ) F*(cu). As such derivations d o not present any special problems, it is left to the student as an exercise to make a systematic r T A B L E II Corresponding properties o f f(t) and F(w) fit) F(co) Real and even real and even Real and o d d imaginary and o d d Imaginary and even imaginary and even Imaginary and o d d real and o d d study. e. cos cot and sin cot. Table II summarizes a few examples which can serve as a start for a more extensive study.

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