Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.8 EXERCISES

Evaluate(d/dx)sinh−^1 xusing the logarithmic form of the inverse.

From the results of section 3.7.5,

d

dx

(

sinh−^1 x

)

=

d

dx

[

ln

(

x+

√

x^2 +1

)]

=

1

x+

√

x^2 +1

(

1+

x

√

x^2 +1

)

=

1

x+

√

x^2 +1

(√

x^2 +1+x

√

x^2 +1

)

=

1

√

x^2 +1

.

3.8 Exercises

3.1 Two complex numberszandware given byz= 3+4iandw=2−i.Onan
Argand diagram, plot
(a) z+w,(b)w−z,(c)wz,(d)z/w,
(e) z∗w+w∗z,(f)w^2 ,(g)lnz,(h)(1+z+w)^1 /^2.

3.2 By considering the real and imaginary parts of the producteiθeiφprove the
standard formulae for cos(θ+φ) and sin(θ+φ).
3.3 By writingπ/12 = (π/3)−(π/4) and consideringeiπ/^12 , evaluate cot(π/12).
3.4 Find the locus in the complexz-plane of points that satisfy the following equa-
tions.

(a) z−c=ρ

(

1+it

1 −it

)

,wherecis complex,ρis real andtis a real parameter

that varies in the range−∞<t<∞.

(b)z=a+bt+ct^2 ,inwhichtis a real parameter anda,b,andcare complex

numbers withb/creal.

3.5 Evaluate

(a) Re(exp 2iz), (b) Im(cosh^2 z), (c) (−1+

√

3 i)^1 /^2 ,

(d)|exp(i^1 /^2 )|, (e) exp(i^3 ), (f) Im(2i+3), (g)ii,(h)ln[(

√

3+i)^3 ].

3.6 Find the equations in terms ofxandyof the sets of points in the Argand
diagram that satisfy the following:
(a) Rez^2 =Imz^2 ;
(b) (Imz^2 )/z^2 =−i;
(c) arg[z/(z−1)] =π/2.

3.7 Show that the locus of all pointsz=x+iyin the complex plane that satisfy

|z−ia|=λ|z+ia|,λ> 0 ,
is a circle of radius| 2 λa/(1−λ^2 )|centred on the pointz=ia[(1 +λ^2 )/(1−λ^2 )].
Sketch the circles for a few typical values ofλ, includingλ<1,λ>1andλ=1.
3.8 The two sets of pointsz=a,z=b,z=c,andz=A,z=B,z=Care
the corners of two similar triangles in the Argand diagram. Express in terms of
a, b,... , C