Sine transform

The Fourier sine transform F, is obtainable by replacing the cosine by the sine in these integrals. 

Seed germination bioassay of root exudates. Bioassay results are presented as a 23 week mean for each germination count time (Table III, IV, V, VI). Means were separated by LSD after data normalization by the inverse sine transformation. 

The most spectacular success of the theory in its quasistatic limit is to show how to film atomic motions during a physicochemical process. As is widely known, photographing atomic positions in a liquid can be achieved in static problems by Fourier sine transforming the X-ray diffraction pattern [22]. The situation is particularly simple in atomic liquids, where the well-known Zernicke-Prins formula provides g(r) directly. Can this procedure be transfered to the quasistatic case The answer is yes, although some precautions are necessary. The theoretical recipe is as follows (1) Build the quantity F q)q AS q,x), where F q) = is the sharpening factor ... 

Consider a single-component real process Y(t) in some fixed interval 0 < t < T. Each realization Yx(t) is an ordinary function of t and may be Fourier-transformed in this interval. To avoid complex coefficients we use the sine transform... 

The Poisson equation can also be solved using the discrete sine transform (Zenger and Bader 2004). The discrete sine transform of a two-dimensional function f(x, y) defined on... 

Thus, the Poisson equation may be solved by applying the discrete sine transform, multiplying the result by... 

The Boltzmann superposition principle can be used to show that the storage modulus is related to the sine transform of G t),... 

Use the Boltzmann superposition integral to derive the storage modulus of a viscoelastic liquid as a sine transform of the stress relaxation modulus G(t) [Eq. (7.149) with 6 eq = 0)]. Also derive the loss modulus as a cosine transform of G(t) [Eq. (7.150) with Ggq = 0] for a viscoelastic liquid. 

In the ease of odd functions a Fourier sine transform is used. Thus, if /(x) is an odd funetion, the Fourier transform is... 

Fig. 40. Phase errors a) linear phase error b) nonlinear phase error. — Interferograms /(s) and spectra I v) obtained from a single-sided interferogram (—) and from a double sided one (----- cos transform,------- sine transform). —The dotted line (.. . ) indicates the undis-...

The Fourier transform can be used to solve the Dirichlet problem in the inhnite domain, and the Fourier sine transform can be used in the semi-inhnite domain. The Fourier cosine transform is appropriate for the Neumann problem in the semi-infinite domain. 

The sine transform becomes zero in this case since the sine function is antisymmetric. Therefore, one can write the Fourier transforms in the simpler cosine forms... 

It is seen that c and s are the Fourier cosine and sine transforms of/(f, d, d) at the angular frequency of intersysfem crossing 2Q/5 and p is the total probability of reencounter. 

If fix) is an odd function, its Fourier transform is a Fourier sine transform ... 

Find the one-sided Fourier sine transform of the function ae ... 

The Fourier sine transform of this function yields the swiming speed distribution... 

Necessity of a Wide q Range. Since in the sine transform (4.22) the integration is in principle to be performed from 0 to infinity in q, and since the interference... 

Figure 4.4 Gross radial distribution function g (r) of polystyrene given by the inverse Fourier sine transform of qi q) according to (4.22). (From Schubach et al.1)...

---