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IV - Powers, Exponentials, Logarithms, Trigonometric Functions
For x E JR, the second formula assumes x :/= o. For x E C, we leave to the
reader the effort of finding the values of x to be avoided in one or the other
case. We have
lim tanh x = 1,
x--++oo
lim tanh x = -1
x-+-oo
since e 2x tends to +00 in the first case and to 0 in the second. We have the
same results for cothx and, further,
lim coth x = -00,
x--+-o
lim cothx = +00.
x--++o
The derivatives of the functions cosh and sinh are immediately provided by
the formulae
tanh'x
coth'x
1/ cosh 2 X = 1 - tanh 2 x,
-1/ sinh2 x = 1 - coth 2 x,
whence their directions of increase: the function tanh increases strictly on -1
to +1 between -00 and +00, the function coth x decreases strictly on -1 to
-00 between -00 and 0, and from +00 to 1 between 0 and +00. The graphs
of these functions can be found in all the best textbooks.
In what concerns the expansions in power series of the functions tanh and
coth, the situation is necessarily the same as for the functions tan and cot:
Newton would have explained to you how to calculate the first terms, but the
general formula is not obvious and involves the Bernoulli numbers which will
appear in Chap. VI. Since moreover the change of variable x = iy transforms
the circular functions into the hyperbolic functions, it is not worth doing the
same calculations twice.
The preceding functions admit inverse maps, at least partially. The function y = coshx maps [0, +oo[ onto [1, +oo[ by the general theorems of
Chap. III, whence a function
x = arg cosh y : [1,+00[---4 [0,+00[;
it is differentiable at every point y where cosh' y :/= 0, i.e. for y > 1, and then
argcosh' y
1/ cosh' x = 1/ sinh x = 1/(cosh2 x - 1)1/2
whence, since cosh x = y,
(16.3)
argcosh' y = (y2 _ 1)-1/2
for y > 1.
We can deduce a power series expansion for arg cosh y from this. Formula (11.21) shows that
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
For x E JR, the second formula assumes x :/= o. For x E C, we leave to the
reader the effort of finding the values of x to be avoided in one or the other
case. We have
lim tanh x = 1,
x--++oo
lim tanh x = -1
x-+-oo
since e 2x tends to +00 in the first case and to 0 in the second. We have the
same results for cothx and, further,
lim coth x = -00,
x--+-o
lim cothx = +00.
x--++o
The derivatives of the functions cosh and sinh are immediately provided by
the formulae
tanh'x
coth'x
1/ cosh 2 X = 1 - tanh 2 x,
-1/ sinh2 x = 1 - coth 2 x,
whence their directions of increase: the function tanh increases strictly on -1
to +1 between -00 and +00, the function coth x decreases strictly on -1 to
-00 between -00 and 0, and from +00 to 1 between 0 and +00. The graphs
of these functions can be found in all the best textbooks.
In what concerns the expansions in power series of the functions tanh and
coth, the situation is necessarily the same as for the functions tan and cot:
Newton would have explained to you how to calculate the first terms, but the
general formula is not obvious and involves the Bernoulli numbers which will
appear in Chap. VI. Since moreover the change of variable x = iy transforms
the circular functions into the hyperbolic functions, it is not worth doing the
same calculations twice.
The preceding functions admit inverse maps, at least partially. The function y = coshx maps [0, +oo[ onto [1, +oo[ by the general theorems of
Chap. III, whence a function
x = arg cosh y : [1,+00[---4 [0,+00[;
it is differentiable at every point y where cosh' y :/= 0, i.e. for y > 1, and then
argcosh' y
1/ cosh' x = 1/ sinh x = 1/(cosh2 x - 1)1/2
whence, since cosh x = y,
(16.3)
argcosh' y = (y2 _ 1)-1/2
for y > 1.
We can deduce a power series expansion for arg cosh y from this. Formula (11.21) shows that