PRELIMINARY CALCULUS
C
c
r + r
O
Q
r
p + p
P
p
Figure 2.13 The coordinate system described in exercise 2.20.
2.20
2.21
A two-dimensional coordinate system useful for orbit problems is the tangentialpolar coordinate system (gure 2.13). In this system a cur
2.2 INTEGRATION
y
C
d
( + d)
dA
()
O
B
x
Figure 2.9 Finding the area of a sector OBC dened by the curve () and
the radii OB, OC, at angles to the x-axis 1 , 2 respectively.
dA = 12 2 d, as illustrated in gure 2.9, and hence the total area between two
ang
3.6 APPLICATIONS TO DIFFERENTIATION AND INTEGRATION
3.6 Applications to dierentiation and integration
We can use the exponential form of a complex number together with de Moivres
theorem (see section 3.4) to simplify the dierentiation of trigonometric fun
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Express sin 3 and cos 3 in terms of powers of cos and sin .
Using de Moivres theorem,
cos 3 + i sin 3 = (cos + i sin )3
= (cos3 3 cos sin2 ) + i(3 sin cos2 sin3 ).
(3.28)
We can equate the real and imaginary coecie
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
3.7 Hyperbolic functions
The hyperbolic functions are the complex analogues of the trigonometric functions.
The analogy may not be immediately apparent and their denitions may appear
at rst to be somewhat arbitrary
2.4 HINTS AND ANSWERS
2.45
If Jr is the integral
xr exp(x2 ) dx
0
show that
(a) J2r+1 = (r!)/2,
(b) J2r = 2r (2r 1)(2r 3) (5)(3)(1) J0 .
2.46
Find positive constants a, b such that ax sin x bx for 0 x /2. Use
this inequality to nd (to two signicant gures)
2.3 EXERCISES
2.10
The function y(x) is dened by y(x) = (1 + xm )n .
(a) Use the chain rule to show that the rst derivative of y is nmxm1 (1 + xm )n1 .
(b) The binomial expansion (see section 1.5) of (1 + z)n is
(1 + z)n = 1 + nz +
n(n 1) 2
n(n 1) (n r +
3.2 MANIPULATION OF COMPLEX NUMBERS
Im z
iz
z
Re z
z
iz
Figure 3.5 Multiplication of a complex number by 1 and i.
multiply z by a complex number then the argument of the product is the sum
of the argument of z and the argument of the multiplier. Hence mul
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
4
sech1 x
cosh1 x
2
4 x
3
2
1
2
cosh1 x
sech1 x
4
Figure 3.14 Graphs of cosh1 x and sech1 x.
Find a closed-form expression for the inverse hyperbolic function y = tanh1 x.
First we write x as a function of y, i.e.
SERIES AND LIMITS
For a series with an innite number of terms and |r| < 1, we have limN r N = 0,
and the sum tends to the limit
a
S=
.
(4.4)
1r
In (4.1), r = 12 , a = 12 , and so S = 1. For |r| 1, however, the series either diverges
or oscillates.
Conside
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
f(z)
5
4
3
2
1
1
2
3
4 z
Figure 3.1 The function f(z) = z 2 4z + 5.
the rst term is called a real term. The full solution is the sum of a real term
and an imaginary term and is called a complex number. A plot of th
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Verify the relation (d/dx) cosh x = sinh x.
Using the denition of cosh x,
cosh x = 12 (ex + ex ),
and dierentiating directly, we nd
d
(cosh x) = 12 (ex ex )
dx
= sinh x.
Clearly the integrals of the fundamental hy
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
(a) the equalities of corresponding angles, and
(b) the constant ratio of corresponding sides,
in the two triangles.
By noting that any complex quantity can be expressed as
z = |z| exp(i arg z),
deduce that
a(B C)
3.8 EXERCISES
Evaluate (d/dx) sinh1 x using the logarithmic form of the inverse.
From the results of section 3.7.5,
d
d
ln x + x2 + 1
sinh1 x =
dx
dx
x
1
1+
=
x + x2 + 1
x2 + 1
2
x +1+x
1
=
x + x2 + 1
x2 + 1
=
1
.
x2 + 1
3.8 Exercises
3.1
Two complex
PRELIMINARY ALGEBRA
forms an equation which is satised by particular values of x, called the roots of
the equation:
f(x) = an xn + an1 xn1 + + a1 x + a0 = 0.
(1.1)
Here n is an integer > 0, called the degree of both the polynomial and the
equation, and th
PRELIMINARY ALGEBRA
at a value of x for which (x) is also zero. Then the graph of (x) just touches
the x-axis. When this happens the value of x so found is, in fact, a double real
root of the polynomial equation (corresponding to one of the mk in (1.9) ha
2.1 DIFFERENTIATION
f(x)
C
Q
P
+
x
Figure 2.4 Two neighbouring tangents to the curve f(x) whose slopes dier
by . The angular separation of the corresponding radii of the circle of
curvature is also .
point P on the curve f = f(x), with tan = df/dx evalu
1.8 EXERCISES
(a) the sum of the sines of /3 and /6,
(b) the sine of the sum of /3 and /4.
1.8
1.9
The following exercises are based on the half-angle formulae.
(a) Use the fact that sin(/6) = 1/2 to prove that tan(/12) = 2 3.
(b) Use the
result of (a) t
1.4 PARTIAL FRACTIONS
in partial fractions, i.e. to write it as
f(x) =
4x + 2
A1
g(x)
A2
= 2
=
+
+ .
h(x)
x + 3x + 2
(x 1 )n1
(x 2 )n2
(1.43)
The rst question that arises is that of how many terms there should be on
the right-hand side (RHS). Although som
PRELIMINARY ALGEBRA
The prime integers pi are labelled in ascending order, thus p1 = 1, p2 = 2, p5 = 7, etc.
Show that there is no largest prime number.
Assume, on the contrary, that there is a largest prime and let it be pN . Consider now the
number q fo
1.4 PARTIAL FRACTIONS
Ai but linear functions of x, i.e. of the form Bi x + Ci . Thus, in the expansion,
linear terms (rst-degree polynomials) in the denominator have constants (zerodegree polynomials) in their numerators, whilst quadratic terms (second-d
PREFACE TO THE THIRD EDITION
student to determine whether an incorrect answer is due to a misunderstanding
of principles or to a technical error.
The remaining four hundred or so even-numbered exercises have no hints or
answers, outlined or detailed, avai
3.9 HINTS AND ANSWERS
3.27
A closed barrel has as its curved surface the surface obtained by rotating about
the x-axis the part of the curve
y = a[2 cosh(x/a)]
lying in the range b x b, where b < a cosh1 2. Show that the total surface
area, A, of the barr
PRELIMINARY ALGEBRA
the well-known result that the angle subtended by a diameter at any point on a
circle is a right angle.
Taking the diameter to be the line joining Q = (a, 0) and R = (a, 0) and the point P to
be any point on the circle x2 + y 2 = a2 ,
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Im z
z1 + z2
z2
z1
Re z
Figure 3.3 The addition of two complex numbers.
or in component notation
z1 + z2 = (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 ).
The Argand representation of the addition of two complex nu
COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Im z
z = x + iy
y
x
y
Re z
z = x iy
Figure 3.6 The complex conjugate as a mirror image in the real axis.
In the case where z can be written in the form x + iy it is easily veried, by
direct multiplication of the co
PRELIMINARY ALGEBRA
1.11
1.13
1.15
1.17
1.19
1.21
1.23
1.25
1.27
1.29
1.31
1.33
Show that the equation is equivalent to sin(5/2) sin() sin(/2) = 0.
Solutions are 4/5, 2/5, 0, 2/5, 4/5, . The solution = 0 has multiplicity
3.
(a) A circle of radius 5 centre