Define a function
where denotes the greatest integer less than or equal to
, also called the floor function.
Then, define a function
- Draw the graph of
for
and prove that
is periodic with period 1.
- For
, prove
and prove that
is also periodic with period 1.
- Give a formula for
in terms of the floor of
.
- Find a
such that
- For the constant
from part (d), define
Prove that
is periodic with period 1 and that if
, we have,
- The graph of
is as follows:
Then, we prove that
is periodic with period 1.
Proof. We compute to show(where I’ve replaced Apostol’s notation
with
since this less likely to cause confusion, also we use the solution to this exercise in the second line)
- Proof. First, we establish the requested formula,
This was the requested formula. Next, to show
is periodic with period 1, we show
for any
,
Then, using the formula in the first part of (b) we have
Further, from part (a) we know
is periodic with period 1, so
, and we have,
by definition of
. Thus,
is periodic with period
- To express
in terms of of
we compute as follows:
where
since
is an integer by definition, and
has period 1, so
for any integer
. Continuing,
Here, we know the
term in the integral is zero since
for all
since
. Then,
This is the requested expression of
in terms of
.
- Here we compute the integral, using the formula for
we established in part (c), and solve for the requested constant
,
- First we give the proof that
is periodic with period 1.
Proof. We compute,Next, we establish the requested formula. If
, we have
as requested.
Hi, can someone help me understand why Q(1)=0?
it seems like it should be Q(1)= c , and Q(x+1)=Q(x)+c
in 1.e ,
Thanks
got it..never mind