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:

m\left(t\right)=1-exp\left(-bt\right)

(26)

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:

V\left(x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{x\cdot \left[1-exp\left(-bt\right)\right]}{{x}^{2}+{\left(t+h\right)}^{2}}}dt

(27)

Combining the equations (3), (6) and (27) gives:

V\left(u\right)=-i\pi exp\left(-hu\right){\displaystyle \underset{0}{\overset{\infty}{\int}}\left[1-exp\left(-bt\right)\right]exp\left(-ut\right)dt}

(28)

It can be easily found that:

V\left(u\right)=-\frac{i\pi bexp\left(-hu\right)}{u\left(b+u\right)}

(29)

Combining the equations (3), (27) and (29), the following Fourier transform pair is obtained:

{\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{x\cdot \left[1-exp\left(-bt\right)\right]}{{x}^{2}+{\left(t+h\right)}^{2}}}dt\leftarrow FT\to -\frac{i\pi bexp\left(-hu\right)exp\left(-bu\right)}{u\left(b+u\right)}

(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:

V\left(x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{x\cdot erf\left(\sqrt{bt}\right)}{{x}^{2}+{\left(t+h\right)}^{2}}}dt

(31)

Combining the equations (3), (6) and (31) gives:

V\left(u\right)=-i\pi exp\left(-hu\right){\displaystyle \underset{0}{\overset{\infty}{\int}}erf\left(\sqrt{bt}\right)exp\left(-ut\right)dt}

(32)

It is well known (Spiegel 1976, Abramowitz and Stegun 1968) that:

{\displaystyle \underset{0}{\overset{\infty}{\int}}erf\left(\sqrt{bt}\right)exp\left(-ut\right)dt=\frac{\sqrt{b}}{u\sqrt{b+u}}}

(33)

Combining the equations (3), (31), (32) and (33), the following Fourier transform pair is obtained:

{\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{x\cdot erf\left(\sqrt{bt}\right)}{{x}^{2}+{\left(t+h\right)}^{2}}}dt\leftarrow FT\to -i\pi exp\left(-hu\right)\frac{\sqrt{b}}{\sqrt{b+u}}

(34)

The third case is that of *m*(*t*) = *t*^{n}, for 0 < *n* < 2.

According to the equation (2), the expression for *V*(*x*) becomes:

V\left(x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{x{t}^{n}}{{x}^{2}+{\left(t+h\right)}^{2}}}dt

(35)

Combining the equations (3), (6) and (35) gives:

V\left(u\right)=-i\pi exp\left(-hu\right){\displaystyle \underset{0}{\overset{\infty}{\int}}{t}^{n}exp\left(-ut\right)dt}

(36)

It is well known (Abramowitz and Stegun 1968) that:

{\displaystyle \underset{0}{\overset{\infty}{\int}}{t}^{n}exp\left(-ut\right)dt=\frac{\Gamma \left(n+1\right)}{{u}^{n+1}}}

(37)

Γ is the Gamma function.

Combining the equations (3), (35), (36) and (37) the following Fourier transform pair is obtained:

{\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{x{t}^{n}}{{x}^{2}+{\left(t+h\right)}^{2}}}dt\leftarrow FT\to -i\pi exp\left(-hu\right)\frac{\Gamma \left(n+1\right)}{{u}^{n+1}}

(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*).