15.9. EXERCISES 399

Then e−b2F (b)− 0 = − 1

2 e−2b2√π + 1

2√

π , F (b) = − 12 e−b2

+ 12√

πe−b2= 0. You

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

11. You can do the same problem as above using contour integration. For b= 0 it followsfrom Problem 2 on Page 262. For b > 0, use the contour which goes from −a to ato a+ ib to −a+ ib to −a. Then let a→ ∞ and show that the integral of e−z2

overthe vertical parts of this contour converge to 0. Hint: You know from the earlierproblem what happens on the bottom part of the contour. Also for z = x+ ib,e−z2

=

e−(x2−b2+2ixb) = eb2e−x2

(cos(2xb)+ isin(2xb)) .

12. Consider the circle of radius 1 oriented counter clockwise. Evaluate∫

γz−6 cos(z)dz

13. Consider the circle of radius 1 oriented counter clockwise. Evaluate∫

γz−7 cos(z)dz

14. Find∫

∞

02+x2

1+x4 dx.

15. Suppose f is an entire function and that it has no zeros. Show there must exist anentire function g such that f (z) = eg(z). Hint: Letting γ (0,z) be the line segmentwhich goes from 0 to z, let ĝ(z)≡

∫γ(0,z)

f ′(w)f (w) dw. Then show that ĝ′ (z) = f ′(z)

f (z) . Then(e−ĝ(z) f (z)

)′= e−ĝ(z)− f ′(z)

f (z) f (z)+ e−ĝ(z) f ′ (t) = 0. Now when you have an entirefunction whose derivative is 0, it must be a constant. Modify ĝ(z) to make f (z) =eg(z).

16. Let f be an entire function with zeros {α1, · · · ,αn} listed according to multiplicity.Thus you might have repeats in this list. Show that there is an analytic functiong(z) such that for all z ∈ C, f (z) = ∏

nk=1 (z−αk)eg(z) Hint: You know f (z) =

∏nk=1 (z−αk)h(z) where h(z) has no zeros. To see this, note that near α1, f (z) =

a1 (z−α1)+a2 (z−α1)2 + · · · and so f (z) = (z−α1) f1 (z) where f1 (z) ̸= 0 at α1.

Now do the same for f1 and continue till fn = h. Now use the above problem.

17. Let F (s) = 2(s−1)2+4

so it is the Laplace transform of some f (t). Use the method of

residues to determine f (t).

18. This problem is about finding the fundamental matrix for a system of ordinary dif-ferential equations Φ′ (t) = AΦ(t) , Φ(0) = I having constant coefficients. Here Ais an n× n matrix and I is the identity matrix. A matrix, Φ(t) satisfying the aboveis called a fundamental matrix for A. In the following, s will be large, larger than allpoles of (sI−A)−1.

(a) Show that L(∫ (·)

0 f (u)du)(s) = 1

s F (s) where F (s)≡L ( f )(s)

(b) Show that L (I) = 1s I where I is the identity matrix.

(c) Show that there exists an n×n matrix Φ(t) such that L (Φ)(s) = (sI−A)−1 .Hint: From linear algebra

((sI−A)−1

)i j=

cof(sI−A) ji

det(sI−A)