- Find the following improper integral. ∫
_{−∞}^{∞}dx Hint: Use upper semicircle contour and consider instead ∫_{−∞}^{∞}dx. This is because the integral over the semicircle will converge to 0 as R →∞ if you have e^{iz}but this won’t happen if you use cosz because cosz will be unbounded. Just write down and check and you will see why this happens. Thus you should useand take real part. I think the standard calculus techniques will not work for this horrible integral. - Find ∫
_{−∞}^{∞}dx. Hint: Do the same as above replacing cosx with e^{ix}. - Consider the following contour.
The small semicircle has radius r and is centered at

. The large semicircle has radius R and is centered at. Use the method of residues to computeThis is called the Cauchy principal value for ∫

_{−∞}^{∞}dx. The integral makes no sense in terms of a real honest integral. The function has a pole on the x axis. Another instance of this was in Problem 6 on Page 1299 where ∫_{0}^{∞}sin∕xdx was determined similarly. However, you can define such a Cauchy principal value. Rather than belabor this issue, I will illustrate with this example. These principal value integrals occur because of cancelation. They depend on a particular way of taking a limit. They are not mathematically respectable but are certainly interesting. They are in that general area of finding something by taking a certain kind of symmetric limit. Such problems include the Lebesgue fundamental theorem of calculus with the symmetric derivative. - Find ∫
_{0}^{2π}dθ. - Find ∫
_{0}^{2π}. - Find ∫
_{−π∕2}^{π∕2}. - Suppose you have a function fwhich is the quotient of two polynomials in which the degree of the top is two less than the degree of the bottom and you consider the contour.
Then define

in which s is real and positive. Explain why the integral makes sense and why the part of it on the semicircle converges to 0 as R →∞. Use this to find

- Show using methods from real analysis that for b ≥ 0,
Hint: Let F

≡∫_{0}^{∞}e^{−x2 }cosdx −e^{−b2 }. Then from Problem 2 on Page 689, F= 0. Using the mean value theorem on difference quotients and the dominated convergence theorem, explain whyThen

You fill in the details. This is meant to be a review of real variable techniques.

- You can do the same problem as above using contour integration. For b = 0 it follows from Problem 2
on Page 689. For b > 0, use the contour which goes from −a to a to a + ib to −a + ib to −a. Then let
a →∞ and show that the integral of e
^{−z2 }over the vertical parts of this contour converge to 0. Hint: You know from the earlier problem what happens on the bottom part of the contour. Also for z = x + ib,e^{−z2 }= e^{−(x2−b2+2ixb) }= e^{b2 }e^{−x2 }. - Consider the circle of radius 1 oriented counter clockwise. Evaluate
- Consider the circle of radius 1 oriented counter clockwise. Evaluate
- Find ∫
_{0}^{∞}dx. - Find ∫
_{0}^{∞}dx - Suppose f is an entire function and that it has no zeros. Show there must exist an entire
function g such that f= e
^{g(z) }. Hint: Letting γbe the line segment which goes from 0 to z, let ĝ≡∫_{γ(0,z) }dw. Then show that ĝ^{′}=. Then^{′}= e^{−ĝ(z) }f+ e^{−ĝ(z) }f^{′}= 0 . Now when you have an entire function whose derivative is 0, it must be a constant. Modify ĝto make f= e^{g(z) }. - Let f be an entire function with zeros listed according to multiplicity. Thus you might have repeats in this list. Show that there is an analytic function gsuch that for all z ∈ ℂ,
Hint: You know f

= ∏_{k=1}^{n}hwhere hhas no zeros. To see this, note that near α_{1},f= a_{1}+ a_{2}^{2}+and so f=f_{1}where f_{1}≠0 at α_{1}. Now do the same for f_{1}and continue till f_{n}= h. Now use the above problem. - Let F=so it is the Laplace transform of some f. Use the method of residues to determine f.
- This problem is about finding the fundamental matrix for a system of ordinary differential
equations
having constant coefficients. Here A is an n×n matrix and I is the identity matrix. A matrix, Φ

satisfying the above is called a fundamental matrix for A. In the following, s will be large, larger than all poles of^{−1}.- Show that ℒ=Fwhere F≡ℒ
- Show that ℒ=I where I is the identity matrix.
- Show that there exists an n × n matrix Φsuch that ℒ=
^{−1}. Hint: From linear algebraShow that the ij

^{th}entry of^{−1}satisfies the conditions of Proposition 16.9.7 and so there exists Φsuch that ℒ=^{−1}. By Corollary 16.9.3, this t → Φis continuous. - Thus ℒ= I. Then explain whyℒ=I = ℒand
and so Φ is a fundamental matrix.

- Next explain why Φ must be unique by showing that if Φis a fundamental matrix, then its Laplace transform must be
^{−1}and use the theorem which says that if the two continuous functions have the same Laplace transform, then they are the same function.

- Show that ℒ
- In the situation of the above problem, show that there is one and only one solution to the initial value
problem
and it is given by

Hint: Verify that ℒ

= ℒℒ. Thus if x is given by the variation of constants formula just listed, then= x_{0}+ ∫_{0}^{t}Axdu + ∫_{0}^{t}fdu. You could also simply differentiate the variation of constants formula using chain rule and verify it works.

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