ˆ

5. Apply Theorem 35, with F = C , and T = Tn z n with Tn = 0 for

n ≥ 1, and T0 f = f (ω) for every f ∈ E .

256 Solutions

Section 6.4:

1. The series in Nos. 1, 2, and 5 are k-summable. The one in No. 3

converges for k ≥ 1, and hence is k-summable in this case, while for

k < 1, there are in¬nitely many singular directions. The one in No. 6

is 1-summable.

2. Proceed as in the proof of Lemma 10 part (b).

3. Verify directly that f (z; ») ∼1 f (z; ») in |d ’ arg z| < π; to do so,

=ˆ

proceed as in the proof of Theorem 22 (p. 79).

∞

m’1

ˆ “(» + n) z n + z m 0 “(» + m + n) z n , for

4. Observe f (z; ») = 0

su¬ciently large integer m, and use the previous exercise.

ˆˆ ˆˆ

5. Let g = S (B1 f ), g(·; ±, β) = S (B1 f (·; ±, β)). For Re β > Re ± > 0,

use the Beta Integral (p. 229) to show

1

“(β)

(1 ’ x)β’±’1 x±’1 g(xu) dx.

g(u; ±, β) =

“(±) “(β ’ ±) 0

ˆ

Use this to conclude f (z; ±, β) ∈ E {z}1 . To remove the restrictions on

ˆ

± and β, use termwise integration resp. di¬erentiation of f (z; ±, β).

Section 6.5:

1. Observe that the geometric series is contained in C {z}, for every

k > 0.

2. Assume f (z) ∼1/k f (z) in a sectorial region of opening larger than

ˆ

=

π/k, and let g = Lk (T ’ f ), g = Lk (T ’ f ). Conclude that g(z) ∼1/k

ˆˆˆ

ˆ =

g (z) in another sectorial region of opening larger than π/k. This shows

ˆ

that (“(1+n/k)/m(n)) is a summability factor; the other case follows

similarly.

3. Follows from the previous exercise.

4. Use the previous exercise and Theorem 31 (p. 94).

5. Verify that f (z) ∼1/k f (z) in G implies f (»z) ∼1/k f (»z) in a corre-

ˆ ˆ

= =

˜

sponding sectorial region G of the same opening.

p’1

ˆ ˆ

fn z n , observe f (z µj ) =

6. For µ = exp[2πi/p] and f (z) = j=0

p fpn z pn , and use the previous exercise.

7. Use termwise integration resp. di¬erentiation!

8. Use Exercise 1, and compare Exercise 1 on p. 99.

9. Use Theorem 37 (p. 106).

Solutions 257

Section 6.6:

4. Letting rn (z) denote the said Laplace transform, show ¬rst

ω

rn+1 (z) = rn (z) ’ n ≥ 1.

rn (zω/[z + ω]),

z+ω

6. Show ¬rst that a power series in powers of z p belongs to E {z}k,d if

and only if it is in E {z}k,dj , for dj = d + 2jπ/p.

Section 7.1:

2. Show that it su¬ces to restrict to |a| < µ.

Section 7.3:

1. Use Theorem 34 (p. 103), recalling that the notion of singular rays

implies that d0 + 2kπ are also singular, for every k ∈ Z.

ˆ “(1 + n/(2k)) z n + “(1 + n/(2k)) (’1)n z n .

2. 2 f (z) =

Section 7.4:

2. For large z and suitable path of integration, write the integral in the

form

z

e’p(u)’» log u (p (u) + »/u) g (1/u) du,

» p(z)

fj (1/z) = z e ˜

∞(2jπ/r)

with a function g analytic near the origin. Then, verify that one

˜

can always choose a path of integration along which the equation

t = p(u) + » log u has a unique solution u = ψ(t). Substitute the

integral accordingly and estimate.

4. Use Proposition 18.

Section 8.1:

(m) (m)

¤ K n for every n, m ≥ 1 (with suitably large

1. Recall that tn , bn

K > 0).

˜ ˜

2. (1 ’ bx) dm+1 (x) = bx (dm (x) + dm (x)), (1 ’ ax) dm+1 (x) = ax dm (x),

for every m ≥ 0.

258 Solutions

Section 8.2:

1. Use Proposition 26 (p. 217).

2. Use the previous exercise.

3. Compare Proposition 13 (p. 105).

Section 8.3:

1. Set n = n0 and use Lemma 24 (p. 212) to conclude Tn0 = 0; then

increase n0 . For r = 0, A0 +n I and B0 have to have disjoint spectrum,

for every n ≥ n0 .

ˆ

2. Block rows, resp. columns, of T (z) according to the block structure

ˆ ˆ

of A(z), resp. B(z). Then, use the previous exercise and invertibility

ˆ

of T (z) to show that to every j there corresponds a unique k so that

˜ ˆ

»j = »k . Permuting columns of T (z), conclude that one may assume

ˆ

this to happen for k = j, so T (z) diagonally blocked. Use invertibility

once more to conclude that then sk = sk . For the convergence proof,

˜

use Exercise 3 on p. 128.

ˆ ˆ

3. Permute rows and columns of B(z), or of A(z), so that the leading

terms agree, which can be done according to the previous exercise.

Then in particular both matrices have the same block structure.

Section 8.4:

1. Observe that, owing to commutativity, (3.5) coincides with the re-

cursion formula for Tn .

2. Show that it su¬ces to consider Λ = diag [»1 Is1 , . . . , »µ Isµ ] with

distinct »m , and show that then N , hence: A, is diagonally blocked

in the block structure of Λ. Next, use Lemma 24 (p. 212) to show that

B commutes with A if and only if it is likewise diagonally blocked,

so B certainly commutes with Λ.

3. Use the series representation. In particular, conclude that T (z) com-

ˆ

mutes with A(z).

4. Use the ¬rst exercise to reduce the problem to one where An = 0 for

n ≥ r + 1. Then, use a constant transformation to put A0 into Jordan

canonical form. Next, use T (z) as in the previous exercise to remove

its nilpotent part, and conclude from above that then all other coef-

¬cients are diagonally blocked in the block structure of A0 . Thus, it

su¬ces to continue with one such block. Repeating these arguments,

conclude that every elementary system (3.3) can be transformed into

Solutions 259

one where the coe¬cient matrix has the desired form, except for the

conditions upon Re bm (0). This can then be arranged using shearing

transformations.

ˆ

6. First, show that for a scalar formal meromorphic series t(z) the

ˆ ˆ

equation z t (z) = constant implies t(z) = constant. Then compute

ˆ ˆ

N1 T (z) ’ T (z) N2 .

Section 8.5:

1. Build a vector x by rearranging the elements of X.

2. Holomorphicity follows from the results in Chapter 1. For the de-

terminant, show that for an arbitrary fundamental solution X2 (z) of

z x = A2 (z) x, the matrix X(z) X2 (z) is a solution of z x = A1 (z) x,

and use Proposition 1 (p. 6).

3. Use the previous exercise.

Section 9.1:

2. For a matrix C = Ejm with a 1 in position (j, m) and 0™s elsewhere,

’1

use the previous exercise to see that X1 (z) Ejm X2 (z) is a solution.

Check that all these solutions are linearly independent, and use The-

orem 2 (p. 6).

4. For (d), show (n, m) ∈ Suppj,n1 if and only if (9.2) holds in Sj =

Sj © . . . © Sj+n1 . Use this to conclude that C has the required form if

and only if Y (z) (I + C) Y ’1 (z) ∼1/k I in Sj . Then, use results from

=

Section 4.5. For (g), enumerate the data pairs in such a way that (9.2)

holds for dj’1 + π/(2k) < arg z < dj + π/(2k) if and only if n < m,

so that Gj,n1 consists of upper triangularly blocked matrices. Then,

consider blocks directly above the diagonal ¬rst, next, treat those in

the next superdiagonal, etc.

Section 9.2:

1. Consider the identities obtained for the blocks of D, C+ , and C’ in

a suitable order.

2. For the product on the right, ¬rst consider the blocks directly below

the diagonal, then in the next subdiagonal, and so forth. Also compare

this to Exercise 4 on p. 142.

260 Solutions

3. Observe that for integer values of k the number j0 always is even.

Then show that, after a suitable renumeration of the data pairs, i.e.,

a permutation of the columns of the HLFFS, we may assume that

(9.2) holds in the sector „j < arg z < „j+1 , if and only if n < m.

Use this to conclude than then the Vj+1 , . . . , Vj+j0 /2 are lower, the

Vj+j0 /2+1 , . . . , Vj+j0 upper triangular, and use the previous exercises.

Finally, use (9.4) to compute the remaining matrices.

Section 9.3:

1. Observe z Y (z) = B(z) Y (z), and use partial integration of (9.6).

2. Estimate analogously to the proof of Theorem 23 (p. 80).

Section 9.4:

2. For z0 ∈ Sj—(k) , observe the previous exercise to invert the integral

representation of the auxiliary functions. Then, deform the path of

integration in Borel™s transform. Finally, check that the integral does

not depend upon the choice of z0 .

Section 9.5:

2. Using Exercise 2 on p. 58, one can compute ¦1 (u; s; k), which is a

2-vector, with second component c (»1 ’ »2 )’1 (u + »2 )1’a’s F (1 +

±, 1 + β; 2 ’ a ’ s; z)/“(2 ’ a ’ s), with z = (u + »1 )/(»1 ’ »2 ). Using

(k)

Exercise 6 on p. 27, one can then compute C21 . Assuming 2 ≺ 1, and

choosing arg(»1 ’ »2 ) in dependence of k accordingly, one ¬nds

2πi (»1 ’ »2 )d’a

(k)

C21 = c.

“(1 + ±) “(1 + β)

Section 9.6:

2. Verify that the spectrum of A0 is directly related to the data pairs of

HLFFS.

Section 9.7:

n+p n+p

2. Show 0 ¤ + xm ) ’ 1 ¤ exp[ xm ] ’ 1, for n, p ∈ N.

m=n (1 m=n

3. Proceed by induction with respect to m.

4. For n, p ∈ N, observe

Tn+p (z) ’ Tn (z) = [I + Tn (z)] {[I + Un+p (z) · . . . · [I + Un+1 (z)] ’ I},

and then estimate as in the previous exercises.

Solutions 261

Section 10.1:

1. Use the de¬nition and justify interchanging the order of integration.

2. Let the jth integral be performed along arg z = dj ; then we may

choose dj’1 so that 2|dj’1 ’ dj | ¤ κ’1 .

j