PROBLEM SET VII
“FUNDAMENTALS”
DUE FRIDAY, 4 NOVEMBER
In this problem set, we will use define some important functions using integrals. Use only the definitions given
here in the following exercises.
R by the following formula:
Zx
1
arctan(x) =
dt .
2
0 1+ t
Definition. Define a function arctan: R
Exercise 47. Show that arctan is increasing and that for any x ∈ R, one has
arctan(−x) = − arctan(x)
and
| arctan(x)| ≤ |x|.
Definition. Now define π as the real number 4 arctan(1).
Exercise 48. Show that the image arctan(R) is the open interval (−π/2, π/2). [Hint: show that for any " > 0, there
exists a real number N > 0 such that for any x > N , one has π/2 − " < arctan(x) < π/2.]
(−π/2, π/2) is a bijection of class C ∞ .
Exercise 49. Prove that arctan: R
Definition. Denote by R+ the ray (0, +∞). Define a function log: R+
Zx
1
log(x) =
dt .
1 t
R by the formula
Exercise 50. Prove that for any x, y ∈ R+ , one has
log(x y) = log(x) + log(y).
Exercise 51. Show that log is increasing, and show that the image log(R+ ) is R.
Exercise 52. Prove that log: R+
R is a bijection of class C ∞ .
Exercise 53. Prove that the inverse exp: R
elements x, y ∈ R,
exp0 (x) = exp(x),
R+ of the function log is a differentiable function such that for any
exp(0) = 1,
and
1
exp(x + y) = exp(x) exp(y).