The case of an infinitely big T represents a fault with a very big vertical dimension. The function V(x), according to the equation (2), may be defined if polarization m is zero for t equal to zero. Three forms of the function m(t) are considered.
In the first case, m(t) is given by:
with b > 0.
A polarization m(t) of the type of equation (26) ensures that the potential V(x) takes finite values. From the physical point of view it expresses a fault of geothermal activity with an increasing polarization with depth, which tends to a constant value. In Figure 2 the variation of m with t is presented.
Equation (2) becomes:
(27)
Combining the equations (3), (6) and (27) gives:
(28)
It can be easily found that:
(29)
Combining the equations (3), (27) and (29), the following Fourier transform pair is obtained:
(30)
The function m(t) = 1-exp(-bt) is equal to zero for t = 0 and increases with t tending to unity.
The function m(t) = erf[sqrt(bt)] (erf is the error function) has a similar behaviour with t, since it is zero for t = 0 and tends to unity as long as t increases. It also ensures finite values of V(x). Taking into account the equation (2), the expression for V(x) becomes:
(31)
Combining the equations (3), (6) and (31) gives:
(32)
It is well known (Spiegel 1976, Abramowitz and Stegun 1968) that:
(33)
Combining the equations (3), (31), (32) and (33), the following Fourier transform pair is obtained:
(34)
The third case is that of m(t) = tn, for 0 < n < 2.
According to the equation (2), the expression for V(x) becomes:
(35)
Combining the equations (3), (6) and (35) gives:
(36)
It is well known (Abramowitz and Stegun 1968) that:
(37)
Γ is the Gamma function.
Combining the equations (3), (35), (36) and (37) the following Fourier transform pair is obtained:
(38)
Equation (38) is valid for 0 < n < 2. For n greater than or equal to 2 the generalised integral of equation (2) does not converge to a finite number.
It is important to mention that the generalized integrals on the right side of equations (28), (32) and (36) are the Laplace transforms of the respective polarization functions m(t).