25.9 EXERCISES
imaginaryz-axes, find the strengths of the field (a) at a point one metre directly
above the fence, (b) at ground level one metre to the side of the fence, and (c)
at a point that is level with the top of the fence but one metre to the side of it.
What is the direction of the field in case (c)?
25.3 For the function
f(z)=ln(
z+c
z−c)
,
wherecis real, show that the real partuoffis constant on a circle of radius
ccosechucentred on the pointz=ccothu. Use this result to show that the
electrical capacitance per unit length of two parallel cylinders of radiia,placed
with their axes 2dapart, is proportional to [cosh−^1 (d/a)]−^1.
25.4 Find a complex potential in thez-plane appropriate to a physical situation in
which the half-planex>0,y= 0 has zero potential and the half-planex<0,
y= 0 has potentialV.
By making the transformationw=a(z+z−^1 )/2, withareal and positive, find
the electrostatic potential associated with the half-planer>a,s=0andthe
half-planer<−a,s= 0 at potentials 0 andV, respectively.
25.5 By considering in turn the transformations
z=^12 c(w+w−^1 )andw=expζ,
wherez=x+iy,w=rexpiθ,ζ=ξ+iηandcis a real positive constant, show
thatz=ccoshζmaps the stripξ≥0, 0≤η≤ 2 π, onto the wholez-plane. Which
curves in thez-plane correspond to the linesξ=constantandη=constant?
Identify those corresponding toξ=0,η=0andη=2π.
The electric potentialφof a charged conducting strip−c≤x≤c,y=0,
satisfies
φ∼−kln(x^2 +y^2 )^1 /^2 for large values of (x^2 +y^2 )^1 /^2 ,
withφconstant on the strip. Show thatφ=Re[−kcosh−^1 (z/c)] and that the
magnitude of the electric field near the strip isk(c^2 −x^2 )−^1 /^2.
25.6 For the equation 8z^3 +z+1=0:
(a) show that all three roots lie between the circles|z|=3/8and|z|=5/8;
(b) find the approximate location of the real root, and hence deduce that the
complex ones lie in the first and fourth quadrants and have moduli greater
than 0.5.25.7 Use contour integration to answer the following questions about the complex
zeros of a polynomial equation.
(a) Prove thatz^8 +3z^3 +7z+ 5 has two zeros in the first quadrant.
(b) Find in which quadrants the zeros of 2z^3 +7z^2 +10z+ 6 lie. Try to locate
them.
25.8 The following is a method of determining the number of zeros of annth-degree
polynomialf(z) inside the contourCgiven by|z|=R:
(a) putz=R(1 +it)/(1−it), witht=tan(θ/2), in the range−∞ ≤t≤∞;
(b) obtainf(z)as
A(t)+iB(t)
(1−it)n
(1 +it)n
(1 +it)n;
(c) it follows that argf(z)=tan−^1 (B/A)+ntan−^1 t;
(d) and that ∆C[argf(z)] = ∆C[tan−^1 (B/A)] +nπ;
(e) determine ∆C[tan−^1 (B/A)] by evaluating tan−^1 (B/A)att=±∞and finding
the discontinuities inB/Aby inspection or using a sketch graph.