The effect of the Kerr gravitational field on wave phenomena is explored by examining the inhomogeneous wave equation for a scalar massive field in a Kerr background geometry. The equation is separated in Boyer-Lindquist coordinates. The angular functions a r e spheroidal harmonics, and the radial equation is reduced to a one-dimensional SchrGdinger equation with an effective potential.

The analytic properties of the spin-weighted angular spheroidal functions introduced by Teukolsky are investigated by means of a series involving Jacobi polynomials. This approach facilitates the numerical determination of eigenvalues, particularly in the case of complex frequencies.

The Chandrasekhar–Page angular functions for the Dirac equation in the Kerr–Newman background are expanded as series of hypergeometric polynomials, and a three-term recurrence relation is derived for the coefficients in these series. This leads to a transcendental equation for the determination of the separation constant which is obtained initially as a power series and is then iterated by the method of Blanch and Bouwkamp.

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The representation of the shape of a biconcave erythrocyte by a set of three parametric equations was achieved by using the expressions that transform the curvilinear coordinates from the disc-cyclide coordinate system [denoted J2R; Moon and Spencer (1988), Field Theory Handbook, Springer-Verlag, Berlin] to Cartesian coordinates. The equations are products of elliptic functions, so the challenge was to relate the three major 'shape-defining' measurements of the human erythrocyte in Cartesian coordinates to three parameters in the new curvilinear coordinates, to give a realistic representation of the shape of the membrane-surface. The relationships between the coefficients of the Cartesian degree-4 surface that describes the discocyte and the coordinate transformation equations were derived with the aid of Mathematica; and the membrane-surface of the cell was drawn using the ParametricPlot3D function in this 'package'. By having the erythrocyte shape expressed in its new form it is readily amenable to further transformations that might be used to model those changes in shape that are seen when the cells are immersed in media of various osmolalities, or when they change metabolic 'states'. On the other hand, the relationship between the coefficients of the Cartesian expression for the disc-cyclide surface is relevant to image analysis of erythrocytes, as determined by physical methods that rely on Cartesian imaging 'slices'. These methods include confocal microscopy and various nuclear magnetic resonance microimaging procedures.