Mathematical Methods for Physics and Engineering - Matematica.NET

25.3 LOCATION OF ZEROSφ =0yπ/αz 0w = z α sw 0φ =0xΦ=0Φ=0r(a)(b)w ∗ 0Figure 25.4 (a) An infinite conducting wedge with interior angle π/α and aline charge at z = z 0 ; (b) after the transformation w = z α , with an additionalimagechargeplacedatw = w0 ∗.Substituting w = z α into the above shows that the required complex potential in theoriginal z-plane isf(z) =q ( ) z α − z0∗αln. ◭2πɛ 0 z α − z0αIt should be noted that the appearance of a complex conjugate in the finalexpression is not in conflict with the general requirement that the complexpotential be analytic. It is z ∗ that must not appear; here, z0 ∗α is no more than aparameter of the problem.25.3 Location of zerosThe residue theorem, relating the value of a closed contour integral to the sum ofthe residues at the poles enclosed by the contour, was discussed in the previouschapter. One important practical use of an extension to the theorem is that oflocating the zeros of functions of a complex variable. The location of such zeroshas a particular application in electrical network and general oscillation theory,since the complex zeros of certain functions (usually polynomials) give the systemparameters (usually frequencies) at which system instabilities occur. As the basisof a method for locating these zeros we next prove three important theorems.(i) If f(z) has poles as its only singularities inside a closed contour C and isnot zero at any point on C then∮f ′ (z)C f(z) dz =2πi ∑ (N j − P j ). (25.14)jHere N j is the order of the jth zero of f(z) enclosed by C. Similarly P j is theorder of the jth pole of f(z) inside C.To prove this we note that, at each position z j , f(z) can be written asf(z) =(z − z j ) mj φ(z), (25.15)879