Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25 Applications of complex variables In chapter 24, we developed the basic theory of the functions of a complex variable,z=x+iy, ...

APPLICATIONS OF COMPLEX VARIABLES y x Figure 25.1 The equipotentials (dashed circles) and field lines (solid lines) for a line c ...

25.1 COMPLEX POTENTIALS the field produced by the line charge. The minus sign is needed in (25.1) because the value ofφmust decr ...

APPLICATIONS OF COMPLEX VARIABLES is described by the complex potential f(z)=kln(z−z 0 ), wherekis the strength of the source. A ...

25.1 COMPLEX POTENTIALS P Q y x nˆ Figure 25.2 A curve joining the pointsPandQ. Also shown isˆn, the unit vector normal to the c ...

APPLICATIONS OF COMPLEX VARIABLES
A conducting circular cylinder of radiusais placed with its centre line passing through the o ...

25.2 APPLICATIONS OF CONFORMAL TRANSFORMATIONS y s s x r r (a)z-plane (b)w-plane (c)w-plane Figure 25.3 The equipotential lines ...

APPLICATIONS OF COMPLEX VARIABLES deduce that F(w)=f(z)=V+ikz=V+ikw^1 /^2 (25.9) is the required potential. Expressed in terms o ...

25.3 Location of zeros φ=0 φ=0 Φ=0 Φ=0 y x π/α z 0 w=zα s r w 0 (a) (b) w^0 ∗ Figure 25.4 (a) An infinite conducting wedge with ...

APPLICATIONS OF COMPLEX VARIABLES whereφ(z) is analytic and non-zero atz=zjandmjis positive for a zero and negative for a pole. ...

25.3 LOCATION OF ZEROS polynomials, only the behaviour of a single term in the function need be con- sidered if the contour is c ...

APPLICATIONS OF COMPLEX VARIABLES Y y X R O x Figure 25.5 A contour for locating the zeros of a polynomial that occur in the fir ...

25.4 SUMMATION OF SERIES
By considering ∮ C πcotπz (a+z)^2 dz, whereais not an integer andCis a circle of large radius, evaluat ...

APPLICATIONS OF COMPLEX VARIABLES Ims Res L λ Figure 25.6 The integration path of the inverse Laplace transform is along the inf ...

25.5 INVERSE LAPLACE TRANSFORM Γ R ΓΓR R L L L (a) (b) (c) Figure 25.7 Some contour completions for the integration pathLof the ...

APPLICATIONS OF COMPLEX VARIABLES Ideally, we would like the contribution to the integral from the circular arc Γ to tend to zer ...

25.5 INVERSE LAPLACE TRANSFORM a b 1 f(x) x Figure 25.8 The result of the Laplace inversion of ̄f(s)=s−^1 (e−as−e−bs)with b>a ...

APPLICATIONS OF COMPLEX VARIABLES 25.6 Stokes’ equation and Airy integrals Much of the analysis of situations occurring in physi ...

25.6 STOKES’ EQUATION AND AIRY INTEGRALS (a) (a) (b) (b) (c) (c) y z Figure 25.9 Behaviour of the solutionsy(z)ofStokes’equation ...

APPLICATIONS OF COMPLEX VARIABLES forward, in that, whatever the sign ofyat any particular pointz, the curvature always has the ...