§2. Series expansions
389
curve x 2 /a 2 + y2/b 2 = 1, i.e. an ellipse with centre O. The parabola y = 3x 2
is parametrised by the map (t, 3t 2 ) of IR into IR 2 •
To parametrise the equilateral hyperbola x 2 - y2 = 1, we need functions
such that X(t)2 - y(t)2 = 1. This equation can also be written as. x(t)2 +
[iy(t)F = 1, so one should choose x(t) = cost, y(t) = isint. This brings us
into C2, but not into 1R 2 if t is real. If, on the other hand, t is pure imaginary,
it is clear from the series expansions that cos t and i sin t become real. One is
thus led to introduce the functions
(16.1 )
cosh t
(16.2)
sinht
cos it = (e t + e- t )/2 =
1 + t 2 /2! + t 4 /4! + ... = 'L)[2n 1 ,
-isinit = (e t - e- t )/2 =
t + t 3 /3! + t 5 /5! + ... = 'L)[2n+11.
These hyperbolic cosine and sine, as one calls them, have properties analogous to those of the circular functions, so similar that their statements hardly
require formal proofs; in fact, one could even deduce them from the properties
of circular functions from the formulae (1) and (2), clearly valid for t E C.
cosh(-x) = cosh x,
cosh 0 = 1,
cosh' x = sinh x,
sinh(-x) = -sinhx.
sinhO=O.
sinh' x = cosh x.
cosh 2 X - sinh2 x = 1.
cosh(x + y) = coshx. coshy + sinhx. sinh y.
sinh(x + y) = sinhx. coshy + sinhy. coshx.
The difference from the circular functions relates to their behaviour for x real.
We have
whence cosh x ~ 1. It is obvious that sinh x is > 0 for x > 0 and < 0 for
x < o. The function sinh x is strictly increasing since sinh' x ~ 1. The function
cosh x is strictly increasing for x > 0 and decreasing for x < o.
As x -+ +00, we clearly have coshx rv eX /2, with the same result for
sinh x; the two functions increase at an exponential rate. As x -+ -00, we
have coshx rv e- x /2 and sinhx rv _e- x /2, so that coshx tends to +00 and
sinh x to -00.
We can pursue the analogy by introducing the functions
tanh x
cothx
sinh x/ cosh x = (e 2x - 1)/(e 2X + 1) = 1 - 2/(e 2x + 1) = -i tan ix,
coshx/sinhx = (e 2X + 1)/(e 2X -1) = 1 +2/(e 2X -1) = icotix.
389
curve x 2 /a 2 + y2/b 2 = 1, i.e. an ellipse with centre O. The parabola y = 3x 2
is parametrised by the map (t, 3t 2 ) of IR into IR 2 •
To parametrise the equilateral hyperbola x 2 - y2 = 1, we need functions
such that X(t)2 - y(t)2 = 1. This equation can also be written as. x(t)2 +
[iy(t)F = 1, so one should choose x(t) = cost, y(t) = isint. This brings us
into C2, but not into 1R 2 if t is real. If, on the other hand, t is pure imaginary,
it is clear from the series expansions that cos t and i sin t become real. One is
thus led to introduce the functions
(16.1 )
cosh t
(16.2)
sinht
cos it = (e t + e- t )/2 =
1 + t 2 /2! + t 4 /4! + ... = 'L)[2n 1 ,
-isinit = (e t - e- t )/2 =
t + t 3 /3! + t 5 /5! + ... = 'L)[2n+11.
These hyperbolic cosine and sine, as one calls them, have properties analogous to those of the circular functions, so similar that their statements hardly
require formal proofs; in fact, one could even deduce them from the properties
of circular functions from the formulae (1) and (2), clearly valid for t E C.
cosh(-x) = cosh x,
cosh 0 = 1,
cosh' x = sinh x,
sinh(-x) = -sinhx.
sinhO=O.
sinh' x = cosh x.
cosh 2 X - sinh2 x = 1.
cosh(x + y) = coshx. coshy + sinhx. sinh y.
sinh(x + y) = sinhx. coshy + sinhy. coshx.
The difference from the circular functions relates to their behaviour for x real.
We have
whence cosh x ~ 1. It is obvious that sinh x is > 0 for x > 0 and < 0 for
x < o. The function sinh x is strictly increasing since sinh' x ~ 1. The function
cosh x is strictly increasing for x > 0 and decreasing for x < o.
As x -+ +00, we clearly have coshx rv eX /2, with the same result for
sinh x; the two functions increase at an exponential rate. As x -+ -00, we
have coshx rv e- x /2 and sinhx rv _e- x /2, so that coshx tends to +00 and
sinh x to -00.
We can pursue the analogy by introducing the functions
tanh x
cothx
sinh x/ cosh x = (e 2x - 1)/(e 2X + 1) = 1 - 2/(e 2x + 1) = -i tan ix,
coshx/sinhx = (e 2X + 1)/(e 2X -1) = 1 +2/(e 2X -1) = icotix.