Given constants such that , prove that
for some constants .
Proof. Define constants and by
(Since we know , so these definitions make sense.) Then
Therefore, we may write,
So, to evaluate the integral we have
For the integral on the right, we make the substitution , so . Therefore,
Prove the following integral formulas.
- Guess and prove a general formula based on parts (a) – (c).
- Proof. We use integration by parts with
This gives us
- Proof. We use integration by parts and the result of part (a). Let
This gives us
- Proof. Again, we use integration by parts, and this time part (b). Let
This gives us
Proof. The proof is by induction. We have already established the case (and and ). Assume then that the formula holds for some positive integer . We then consider the integral
Integrating by parts, we let
Therefore, integrating by parts and using the induction hypothesis we have,
Therefore, the formula holds for , and hence, for all positive integers
Let be a function continuous on an interval . The volume of the solid of revolution obtained by rotating about the -axis on the interval is given by
for every . Find a formula for the function .
Using the formula for the volume of the solid of revolution generated by a function on an interval we know
Now we differentiate both sides of this equation using the fundamental theorem of calculus on the right-hand side,
Prove the identity:
Proof. Using the hint (that ) we start with the expression on the left,
(The interchange of the sum and integral is fine since it is a finite sum. Those planning to take analysis should note that this cannot always be done in the case of infinite sums.) Now, we have a reasonable integral, but we still want to get everything back into the form of the sum on the right so we make the substitution , . This gives us new limits of integration from 1 to 0. Therefore, we have
Let be a function which is differentiable everywhere and which satisfies
for some positive constants and . What can you conclude about such a function ?
(Note: I’m not entirely sure what Apostol wants here since the instruction “what can you conclude” is pretty vague. He does give an “answer” in the back of the book, so I verify that it does have the properties indicated, but I don’t know how you would arrive at that expression just from the question statement. I’ll mark this question as incompletely and hopefully come up with something better in the future.)
Since satisfies the functional equation we can write
Thus, is indeed periodic with period and so
So, this definition of in terms of the periodic function indeed satisfies the functional equation.