(^502) Exponential and logarithmic functions
(^18) Derive the formula
fez +1dx=x-log(ex+1)+c
with the aid of the identity
I ez+1 -ex es
ex+1= eZ+I = I ez+I
(^19) Letting F(O) = 0 and F(x) = e-11" when x 0 0, show that the formula
{1) F(k)(x) = Pk() e-11z2X 3k
holds when x 0 0, k = 1, and Po(x) = 2. Supposing that (1) is valid when k is
a given positive integer n and P, (x) is a particular polynomial in x, differentiate
(1) to obtain a formula which shows that (1) holds when k = n + 1 and
(2) (2 - 3nx2)Pn(x) - x3P',(x),
so that P,+i is a polynomial in x. This shows (the principle involved being called
mathematical induction) that (1) holds for each k, Pk being a polynomial in x.
20 Prove that if P and Q are polynomials in x for which Q(x) is not always
zero, then
(1) limP(x)a '1_' = 0.
z-.o Q(x)
Solution: We can determine constants such that bQ 0 0, qis an integer, and
(2)
_ aQ+ a,x + + amxm
() Q(x) bo + bix + + x°
It therefore suffices to prove that
(3) lim jxjge 'I' = 0,
x-4o
or, as we see by setting t = 1/x2,
(4) lim t-e/2e ' = 0.
1-.
But when t > 1 and k is a positive integer for which -q/2 < k,
(5) t-at2et+ < tke '=
j k
e'
Our conclusion is therefore a consequence of the formula
(6) Iimtk= 0
t--, m et
which is proved in Problem 12.
(^21) Let F be the function for which f(0) = 0 and
(1) F(x) = e-11=:
when x ; 0. With the aid of results of the two preceding problems, prove that