ANALYSIS I: PROBLEM SET # 5
DUE FRIDAY, 1 APRIL
Definition. We introduced three power series with an infinite radius of convergence:
∞ 1
X
exp(z) :=
n=0
n!
zn;
∞
X
(−1)n
z 2n+1 ;
(2n
+
1)!
n=0
∞ (−1)n
X
z 2n .
(2n)!
n=0
sin(z) :=
cos(z) :=
Let us define two more power series, analogous to those of sin and cos:
sinh(z) :=
cosh(z) :=
∞
X
n=0
∞
X
1
(2n + 1)!
1
n=0 (2n)!
z 2n+1 ;
z 2n .
Exercise 34. Prove that the power series sinh and cosh have infinite radii of convergence. Show that all the resulting
functions sin: C
C, cos: C
C, sinh: C
C, and cosh: C
C are continuous.
Exercise 35. Verify the following identities for any real numbers x and y.
exp(i x)
=
cos(x) + i sin(x);
exp(x)
=
cosh(x) + sinh(x);
cosh(x + i y)
=
cosh(x) cos(y) + i sinh(x) sin(y);
sinh(x + i y)
=
sinh(x) cos(y) + i cosh(x) sin(y).
Definition. Consider the power series
arctan(z) :=
∞ (−1)n
X
n=0
2n + 1
z 2n+1 .
Exercise 36. Show that the power series arctan has radius of convergence 1. Show that arctan(−1) and arctan(1) each
converge. For which other complex numbers z of modulus 1 can you say something about the convergence of the
series arctan(z)? Define π := 4 arctan(1), and show that arctan defines a homeomorphism [−1, 1]
[−π/4, π/4]
whose inverse is the map given by tan(x) := sin(x)/ cos(x).
Exercise 37. Show that the composite gd(x) := arctan(sinh(x)) defines a continuous bijection
• π π˜
Š
€p
Š—
” €p
− ,
.
gd: log 2 − 1 , log 2 + 1
4 4
This function is called the Gudermannian.
p
p
Exercise 38. Prove that for any real number x with log( 2 − 1) < x < log( 2 + 1), one has
cos(gd(x)) cosh(x) = 1
sin(gd(x)) cosh(x) = sinh(x).
and
1
2
DUE FRIDAY, 1 APRIL
Definition. Let us now define two interesting rational sequences, which will serve as coefficients for some interesting power series.
The Bernoulli numbers B0 , B1 , B2 , . . . are defined recursively in the following manner. Set B0 := 1, and for any
integer m ≥ 1, define B m as the unique rational number so that
m m + 1
X
Bk = 0.
k
k=0
Similarly, the (even) Euler numbers E0 , E2 , E4 , . . . are defined recursively in the following manner. Set E0 := 1, and
for any integer m ≥ 1, define E2m as the unique rational number so that
m 2m X
E2k = 0.
2k
k=0
Exercise 39. Show that the Euler numbers E2m are all integers. Show that not all Bernoulli numbers are integers,
but for any integer m ≥ 1, the Bernoulli number B2m+1 = 0.
Exercise? 40. Show that
v
v
p
u
u
B B u1
u1
2
2n 2n t
t
(2n + 2)(2n + 1) (2n + 2)(2n + 1) .
lim sin
= lim cos
=
n→∞
n→∞
B2n+2 B2n+2 8
8
2
Conclude that the radius of convergence of the power series
∞ B
X
n
n=0
n!
zn
is 2π, where π is as defined above.
Exercise? 41. Compute the radius of convergence of the power series
∞
X
E2n
x 2n+1 ,
(2n
+
1)!
n=0
p
p
and show that it converges to the Gudermannian gd(x) when log( 2 − 1) < x < log( 2 + 1).
OPTIONAL EXERCISES
Definition. Suppose k an integer. Then the k-th polylogarithm is the series
∞ 1
X
Lik (z) :=
zn.
k
n=1 n
Exercise. Compute the radius of convergence of Lik , and note that it is at least 1.
Exercise. Show that if k is a negative integer, then (z − 1)1−k Lik (z) is a polynomial of degree −k with integral
coefficients. What are the roots of this polynomial? What can you say about its coefficients?
Definition. If k is a positive integer, then the generalized harmonic numbers Hn (k) are the partial sums of the
generalized harmonic series:
n 1
X
.
Hn (k) :=
k
j =1 j
Exercise. Show that for any integer n ≥ 2, the generalized harmonic number Hn (k) is not an integer.
Exercise. Compute the radius of convergence of the power series
∞
X
M k (z) :=
Hn (k)z n .
n=1
Show, moreover, that within this radius of convergence, one has
(z − 1)M k (z) = Lik (z).