3.9 HINTS AND ANSWERS
3.27 A closed barrel has as its curved surface the surface obtained by rotating about
thex-axis the part of the curve
y=a[2−cosh(x/a)]lying in the range−b≤x≤b,whereb<acosh−^1 2. Show that the total surface
area,A, of the barrel is given by
A=πa[9a− 8 aexp(−b/a)+aexp(− 2 b/a)− 2 b].3.28 The principal value of the logarithmic function of a complex variable is defined
to have its argument in the range−π<argz≤π.Bywritingz=tanwin terms
of exponentials show that
tan−^1 z=1
2 iln(
1+iz
1 −iz)
.
Use this result to evaluatetan−^1(
2
√
3 − 3 i
7)
.
3.9 Hints and answers3.1 (a) 5 + 3i;(b)− 1 − 5 i; (c) 10+5i;(d)2/5+11i/5; (e) 4; (f) 3− 4 i;
(g) ln 5 +i[tan−^1 (4/3) + 2nπ]; (h)±(2.521 + 0. 595 i).
3.3 Use sinπ/4=cosπ/4=1/
√
2 ,sinπ/3=1/2andcosπ/3=√
3 /2.
cotπ/12 = 2 +√
3.
3.5 (a) exp(− 2 y)cos2x;(b)(sin2ysinh 2x)/2; (c)
√
2exp(πi/3) or√
2exp(4πi/3);
(d) exp(1/√
2) or exp(− 1 /√
2); (e) 0. 540 − 0. 841 i; (f) 8 sin(ln 2) = 5.11;
(g) exp(−π/ 2 − 2 πn); (h) ln 8 +i(6n+1/2)π.
3.7 Starting from|x+iy−ia|=λ|x+iy+ia|, show that the coefficients ofxandy
are equal, and write the equation in the formx^2 +(y−α)^2 =r^2.
3.9 (a) Circles enclosingz=−ia,withλ=expc>1.
(b) The condition is that arg[(z−ia)/(z+ia)] =k. This can be rearranged to give
a(z+z∗)=(a^2 −|z|^2 )tank, which becomes inx, ycoordinates the equation
of a circle with centre (−acotk,0) and radiusacoseck.
3.11 All three conditions are satisfied in 3π/ 2 ≤θ≤ 7 π/4,|z|≤4; area = 2π.
3.13 Denoting exp[2πi/(2m+ 1)] by Ω, expressx^2 m+1−a^2 m+1as a product of factors
like (x−aΩr) and then combine those containing Ωrand Ω^2 m+1−r.Usethefact
that Ω^2 m+1=1.
3.15 The roots are 2^1 /^3 exp(2πni/3) forn=0, 1 ,2; 1± 31 /^4 ;1± 31 /^4 i.
3.17 Consider (1 +i)n.(b)S 2 (n)=2n/^2 sin(nπ/4).S 2 (6) =−8,S 2 (7) =−8,S 2 (8) = 0.
3.19 Use the binomial expansion of (cosθ+isinθ)^4.
3.21 Show that cos 5θ=16c^5 − 20 c^3 +5c,wherec=cosθ, and correspondingly for
sin 5θ.Usecos−^2 θ=1+tan^2 θ. The four required values are
[(5−
√
20)/5]^1 /^2 ,(5−
√
20)^1 /^2 ,[(5+
√
20)/5]^1 /^2 ,(5+
√
20)^1 /^2.
3.23 Reality of the root(s) requiresc^2 +b^2 ≥a^2 anda+b>0. With these conditions,
there are two roots ifa^2 >b^2 , but only one ifb^2 >a^2.
Fora^2 =c^2 +b^2 ,x=^12 ln[(a−b)/(a+b)].
3.25 Reduce the equation to 16 sinh^4 x= 1, yieldingx=± 0 .481.