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1. Transcendency Of The Exponential Functions 
For the definition of transcendental functions, see Section
6.3.1 Definition_3.1. In this section were going to show that the
exponential, logarithmic, hyperbolic, and inverse hyperbolic functions are
transcendental. Like the function sin x as
seen in
Section
6.3.1 Eq. [3.3], these functions have their own infinite expansions in x, which can be used to compute their
approximate values.
Theorem 1.1  Transcendency Of The Natural Exponential Function
The natural exponential function e^{x} is a transcendental function. 
Proof
for all x, where q_{n}(x) isn't identically 0. Dividing both sides of Eq. [2] by e^{x} we get:
EOP
Theorem 1.2  Transcendency Of The General Exponential Function
The general exponential function b^{x} is a transcendental function. 
Proof
where the a_{k}_{,}_{i}'s are the real coefficients and m_{k} is the degree of p_{k}. So:
EOP
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2. Transcendency Of The Logarithmic Functions 
In Section
6.3.2 Theorem 2.1, it's shown that a function is transcendental iff its
inverse function is transcendental. So the
natural logarithm function ln x, which is the
inverse of the transcendental natural exponential function e^{x}, is
transcendental.
There's also a way to prove directly that
the function ln x is transcendental,
ie, by utilizing the definition of transcendental
functions. We're going to do it, believing that it may be of interest to some
students, especially math majors.
The Natural Logarithm Function
Isnt A Rational Function
The direct proof of the transcendency of the natural
logarithm function that we're going to present employs the property that
this function isnt a rational function. Recall that a function f (x) is said to be a rational function if it is or can be written in the
form of the ratio p(x)/q(x),
where p(x)
and q(x)
are polynomials in the real variable x
with real coefficients and q(x) isnt
identically 0.
We say that polynomial D(x) (the dividend) is divisible by the
polynomial d(x)
(the divisor) or that d(x) divides D(x) if the
remainder of the division of D(x) by d(x) is 0, ie, if D(x)/d(x) = Q(x) (the quotient) or D(x) = d(x)Q(x), where Q(x) is a
polynomial, ie, if d(x) is a factor of D(x). If D(x) has a non0 constant term, then D^{2}(x) has a non0 constant term. If D^{2}(x) is
divisible by the polynomial x,
then its constant term is 0:
D^{2}(x) = xQ(x) = a_{n}x^{n} + a_{n}_{1}x^{n}^{1} + a_{n}_{2}x^{n}^{2} + ... + a_{2}x^{2} + a_{1}x,
assuming Q(x) = a_{n}x^{n}^{1} + a_{n}_{1}x^{n}^{2} + a_{n}_{2}x^{n}^{3} +
+ a_{2}x + a_{1}.
So the constant term of D(x) must also be 0, for otherwise the
constant term of D^{2}(x)
would be non0. Thus D(x) is also divisible by x.
Lemma 2.1  The Natural Logarithm Function Isnt A Rational Function
The natural logarithm function ln x isnt a rational function. 
Proof
Assume that ln
x is a rational function, so that ln x = p(x)/q(x), where p(x) and q(x) are polynomials and q(x)
isn't 0 for any
x > 0 (the function ln x is defined for all x
> 0, so p(x)/q(x)
must be defined for all x
> 0). Cancel all the common factors of
p(x)
and q(x)
if any, and reuse the letters p and
q, so that p(x) and q(x) have no common factors other than 1.
Well show
that p(x)
and q(x)
have x as their common factor,
which is a contradiction. Differentiating both sides of ln x
= p(x)/q(x) we
have:
EOP
Transcendency Of The Natural Logarithm
Function
We now prove that the natural logarithm function ln x is transcendental.
Theorem 2.1  Transcendency Of The Natural Logarithm Function
The natural logarithm function ln x is a transcendental function. 
Proof
The last equation shows that ln x
is a rational function, which is impossible by Lemma 2.1. It follows that the
coefficient of
ln^{n}^{1} x in
Eq. [4] isn't identically 0.
We've shown that Eq. [4] with the (ln x)degree
being n 1 satisfies the
assumption that ln
x is an algebraic function. This
contradicts the fact that n is
the minimum. Therefore ln
x must be a transcendental function.
EOP
Transcendency Of The General Logarithmic
Function
Recall that for any constant b
positive and different from 1, we have log_{b}
x = ( ln x)/ ln b = (1/ ln b) ln x.
Of course ln
b is a
non0 constant, and so is 1/ ln b. As seen in Section
6.3.1 Theorem 3.1, for any non0 constant c,
a function f
(x) is
transcendental iff c f (x) is
transcendental. Thus, as ln
x is transcendental, log_{b} x
is also transcendental.
Corollary 2.1  Transcendency Of General Logarithmic Function
The general logarithmic function log_{b} x is a transcendental function. 
It can also be proved directly that the function log_{b} x is transcendental, as carried out in problem & solution 3.
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3. Transcendency Of The Hyperbolic Functions And Their Inverses 
Since the hyperbolic functions are defined in terms of the
natural exponential function which is transcendental, it's only natural
to expect that they may be transcendental too. We're going to show that they
indeed are.
Theorem 3.1  Transcendency Of Sine And Cosine Hyperbolic Functions
The sine and cosine hyperbolic functions sinh x and cosh x are transcendental functions. 
Proof
Substitute these expansions into Eq. [1]. Then for each
term, multiply out its p_{k}(x)
and expansion. Factor out each factor e^{mx}
that occurs twice or more. The power e^{mx} with the greatest m is e^{nx}, and the power e^{mx}
with the least m is e^{}^{nx}.
There are
2n + 1 terms (m goes from n
to 0 to n). Eq. [1] now becomes:
^{{2.1}} Section 6.3.1 Theorem 3.1.
EOP
Theorem 3.2  Transcendency Of Tangent Hyperbolic Function
The tangent hyperbolic function tanh x is a transcendental function. 
Proof
We have:
^{{2.2}} Section 6.3.1 Problem & Solution 5.
^{{2.3}} Section
6.3.2 Theorem 1.1.
^{{2.4}} Section
6.3.1 Theorem 3.1.
EOP
Corollary 3.1 Transcendency Of The Reciprocal Hyperbolic Functions
The cotangent, secant, and cosecant hyperbolic functions coth x, sech x, and csch x are transcendental functions. 
This corollary is a direct consequence of Theorems 3.1 and 3.2 above and Section 6.3.2 Theorem 1.1.
Corollary 3.2 Transcendency Of The Inverse Hyperbolic Functions
The inverse sine, cosine, tangent, cotangent, secant, and
cosecant hyperbolic functions sinh^{1} x, cosh^{1} x, 
This corollary is a direct consequence of Theorems 3.1 and 3.2 and Corollary 3.1 above and Section 6.3.2 Theorem 2.1.
Problems &
Solutions

1. Is the function:
algebraic or transcendental? Justify your answer.
Solution
It's algebraic, because it's a polynomial.
2. Is it possible that:
for all x > 0? Explain, without doing any numerical calculations to compare the 2 sides.
Solution
No, because ln
x is a transcendental function while the
expression on the righthand side is a finite combination of algebraic
operations applied to polynomials in x and
thus is an algebraic function.
3. Prove directly
(ie, by using the definition of a transcendental function) that the general
logarithmic function log_{b} x
is
transcendental.
Solution
4. Prove that the
cosine hyperbolic function cosh
x is a transcendental function. Hint: For
any non0 constant c, a
function f is
transcendental iff the function cf
is transcendental.
Solution
Substitute these expansions into Eq. [1]. Then for each
term, multiply out its p_{k}(x)
and its expansion. Factor out each factor
e^{mx}
that occurs twice or more. The power e^{mx} with the greatest m is e^{nx} and the power e^{mx}
with the least m is e^{}^{nx}.
There are
2n + 1 terms (m goes down from n
to 0 to n). Eq. [1] now becomes:
5. Prove directly
(ie, by using the definition of a transcendental function) that the cotangent
hyperbolic function coth
x is a
transcendental function. Hint: if f (x) is
transcendental then 1 + f (x), cf (x),
where c is any non0 constant, and
1/ f (x)
are
transcendental.
Solution
We have:
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