Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions
6.5. Evaluation of definite integrals
6.5. EVALUATION OF DEFINITE INTEGRALS 113
Proof. Let z : [0,1] → D be a C1-curve with z(t) = 0 for all t. Then
w=g◦z : [0,1]→g(D) is also aC1-curve and
w(t) =gz z(t) +gz z(t) . Sinceg preserves angles, argw(t)
z(t) must be independent of argz(t). But
w(t)
z(t) =gz+gz z(t) z(t)
and therefore gz ≡0.
(5) The change in infinitesimal areas is given by multiplication by
|f(z0)|2.
Proof. We compute the Jacobian of the map f:
J(f) =
ux vy vx vy
=uxvy −uyvx =u2x+vx2 =|f(z0)|2.
−R R
γR
Figure 6.1. The path of integration for Example (1)
the function F has (possibly) nonzero residues only at these four roots of unity and we conclude that
γR
1
z4+ 1 dz = 2πı
Res
F, eπı4 + Res
F, e3πı4
. The residues are easy to compute:
Res(F, eπı4 ) = 1
(z2+ı)(z+eπı4)
z=eπı4
= −1
2(√
2−ı√ 2) and
Res(F, e3πı4 ) = 1
(z2−ı)(z+e3πı4 ) z=e3πı4
= 1
2(√ 2 +ı√
2). Next we estimate the absolute value of the integral over the semicircle {z; z ≥0, |z|=R}:
π
0
R ı eı θ
R4e4ı θ+ 1 dθ
≤ πR
R4−1 →0 as R →+∞. We conclude that
∞
−∞
1
x4 + 1dx=
√2π 2 .
Remark 6.24. This method will work for the evaluation of integrals of the form
+∞
−∞
Q(x) dx, where Q is a rational function with no singularities onR and with ν∞(Q)≥2.
(2) A second class of integrals that can be evaluated by the Residue Theorem consists of those of the form
I = 2π
0
Q(cosθ,sinθ) dθ ,
where Qis a rational function of two variables with no singu- larities on the unit circle S1 ={z;|z|= 1}.
6.5. EVALUATION OF DEFINITE INTEGRALS 115
To apply the Residue Theorem, we expressI as an integral of a holomorphic function over the unit circle. We use the change of variables
z =eiθ, hence dz =eı θı dθ =ı z dθ and
cosθ = eı θ +e−ı θ
2 = z+z−1
2 , sinθ = eı θ−e−ı θ
2ı = z−z−1 2ı . Example 6.25. Let 0< b < a, and evaluate
I = 2π
0
1
a+bcosθ dθ =
|z|=1
1 (ı z)
a+bz+2z1 dz
=
|z|=1
−2ı
bz2+ 2az+bdz
= 2π ı
|z|<1
Res
−2ı
bz2+ 2az +b, z .
The denominator of the integrand in the last integral is a qua- dratic polynomial in z with precisely one root inside the unit circle (the product of the roots is +1). We conclude that
I = 2π(a2−b2)−12.
(3) The last of the types of integrals to be discussed here is I =
∞
−∞
Q(x)eıxdx ,
where Qis a rational function with (at least) a simple zero at infinity and, in general, with no singularities on R.
We illustrate with a more complicated example, where Q has a simple pole at the origin. Here the ordinary integral is replaced by its principal value (pr. v.) defined below.
pr. v.
∞
−∞
eı x
x dx= lim
δ→0+ R1→+∞ R2→+∞
R2 δ
+ −δ
−R1
eı x
x dx .
We must choose a nice contour for integration; start with largeX1, X2, andY and smallδ, all positive. Our closed path γ has several segments (see Figure 6.2):
γ1 : from −X1 to−δ on R,
γ2 : the semicircle in the lower half-plane of radiusδ and center 0,
−X1 X2
X2+ıY
−X1+ıY
γ2
γ1 γ3
γ4
γ5
γ6
Figure 6.2. The path of integration for Example (3) γ3 : fromδ to X2 on R,
γ4 : at x=X2 go up to height Y,
γ5 : at height Y travel from X2 back to−X1, and (finally) γ6 : at x=−X1 go down from height Y to the real axis.
We start with
γ
eı z
z dz = 2πıRes(f,0),
where f(z) = eı zz = 1z +g(z), with g entire. Thus
γ
eı z
z dz = 2π ı . We now estimate the integral over γ4:
Y 0
eı(X2+ı y) X2+ı y ı dy
≤ Y
0
e−y 1
|X2+ı y|dy
≤ 1 X2
Y 0
e−ydy < 1 X2. Next we estimate the integral over γ5:
−X1 X2
eı(x+ı Y) x+ı Y dx
≤ X2
−X1
e−Y
|x+ı Y| dx
≤e−Y X2
−X1
1
Y dx= e−Y
Y [X2+X1].
Also the integral over γ6: 0
Y
eı(X1+ı y) X1+ı y ı dy
< 1 X1 .
EXERCISES 117
We conclude that
γ
eı z
z dz = lim
δ→0+ X1→+∞
X2→∞
γ1∪γ2∪γ3
eı z z dz.
Finally, lim
δ→0+
X2 δ
+ −δ
−X1
eı x x dx
= lim
δ→0+
γ1∪γ2∪γ3
eı z z dz+
γ2−
eı z z dz . But
lim
δ→0+
γ2−
(z−1 +g(z))dz = lim
δ→0+
γ2−
z−1dz
because g is bounded on a neighborhood of 0 and the length of the path of integration goes to zero. Now
lim
δ→0+
γ2−
z−1 dz = lim
δ→0+
−π 0
1
δeı θδeı θ ı dθ =−πı.
We conclude that pr.v.
∞
−∞
eı x
x dx=πı.
Using the fact that eı x = cosx+ısinx, we see that we have evaluated two real integrals:
pr.v.
∞
−∞
cosx
x dx= 0 and ∞
0
sinx
x dx= π 2. Exercises
6.1. Use Rouch´e’s theorem to prove the Fundamental Theorem of Algebra.
6.2. Let g be a holomorphic function on |z| < R, R > 1, with
|g(z)| ≤1 for all |z| ≤R.
(1) Show that for all t∈C with |t|<1, the equation z =tg(z)
has a unique solution z =s(t) in the disk |z|<1.
(2) Show that t → s(t) is a holomorphic function on the disk
|t|<1.
6.3. Verify (6.1) using Laurent series expansions forf and F.
6.4. Evaluate ∞
−∞
x 4 +x4 dx.
6.5. If f is a holomorphic function on 0 < |z| < 1 and f does not assume any value wwith |w−1|<2, what can you conclude?
6.6. Compute ∞
−∞
dx 1 +x6.
6.7. Evaluate the following integrals.
(a) ∞
−∞
(x+ 1) x4+ 1 dx, (b)
π 0
√ dθ
5 + cosθ, (c)
|z|=1
z6dz 7z7−1, (d)
|z−100π|=1992 π
zcotz dz.
6.8. Let f be an entire function such that |f(z)| = 1 for |z| = 1.
Which are the possible values for f(0) and for f(17)?
6.9. Find ∞
0
dx
1 +x3 using residues.
6.10.Find all functionsf which are meromorphic in a neighborhood of {|z| ≤1}and such that|f(z)|= 1 for |z|= 1, f has a double pole at z = 12, a triple zero atz =−13, and no other zeros or poles in{|z|<1}. 6.11. Suppose f is an entire function satisfying f(n) = n5 and f
−n2
= n7 for all n ∈ Z>0. How many zeros does the function g(z) = [f(z)−e][f(z)−π] have?
6.12. Evaluate
|z|=3
f(z) f(z)−1dz, where f(z) = 2−2z+z2+ z3
81.
6.13. Suppose f is holomorphic for |z| < 1 and f1
n
= n73 for n = 2,3, . . . . What can be said about f(0)?
6.14. Let f be an entire function such that |f(z)| ≤ |z|233 for all
|z|>10. Compute f(8)(10.001).
6.15. Evaluate
|z−π2|=3.15ztanz dz.
EXERCISES 119
6.16. Evaluate the following real integrals using residues:
∞
−∞
cosx 1 +x2dx ,
∞
−∞
sinx 1 +x2dx . 6.17. FindallLaurent series of the form∞
−∞anznfor the function f(z) = z2
(1−z)2(1 +z).
6.18. Iff is an entire function such that f(z)>−2 for all z ∈C and f(ı) =ı+ 2, what isf(−ı)?
6.19. If f is holomorphic on 0 < |z| < 2 and satisfies f(n1) = n2 and f(−n1) =n3 for all n∈Z>0, what kind of singularity does f have at 0?
6.20. LetD be an open, bounded, and connected subset of Cwith smooth boundary.
Iff is a nonconstant holomorphic function in a neighborhood of the closure of Dsuch that |f|=cis constant on∂D, show thatf takes on each value a such that |a|< c at least once in D.
6.21. Suppose f is holomorphic in a neighborhood of the closure of the unit disk.
Show that for |z| ≤1
f(z)(1− |z|2) = 1 2πı
|τ|=1
1−zτ¯
τ−z f(τ)dτ and conclude that the following inequality holds:
|f(z)|(1− |z|2)≤ 1 2π
2π
0
f(expı θ) dθ .
6.22. Letf be an entire function. Suppose that|f(z)| ≤A+B|z|10 for all z ∈C. Show that f is a polynomial.
6.23. Suppose f is meromorphic in a neighborhood of the closed unit disk, that it does not have zeroes nor poles in the unit disk, and that |f(z)|= 1 for |z| = 1. Find the most general such function.
6.24. LetC denote the positively oriented unit circle. Consider the function
f(z) = 2z26
81 + exp
z21 z− 1 2
2 z− 1
3
3
. Evaluate the following integrals:
C
f(z)dz;
C
f(z)dz;
C
f(z) f(z) dz.
6.25. If f is entire and satisfies |f(z)−3| ≥ 0.001 for all z ∈ C, f(0) = 0, f(1) = 2, f(−1) = 4, what is f(ı)?.
6.26. If f is holomorphic for 0<|z|<1 andf1
n
=n2, f
−n1
= 2n2 for n= 2,3,4, . . ., what can you say about f?
6.27. Find all series of the form ∞
−∞
anzn that converge in some domain to
f(z) = 2−z2 z(1−z)(2−z).
6.28. Suppose f is entire and f(z) = t2 for all z ∈ C and for all t ∈R. Show that f is constant.
6.29. Find all Laurent series of the form ∞
−∞
anzn representing the function
f(z) = 1
(z−1)(z−2)(z−3). 6.30. Iff is holomorphic for 0<|z|<1,f1
n
=n−2 andf
−12
= 2n−2,n = 2,3,4, . . ., find lim
z→0|f(z)−2|. 6.31. Find
∞
0
sin22x
x2 dx using residues.
6.32. Prove the following extension of the Maximum Modulus Prin- ciple. Let f be holomorphic and bounded on |z| <1, and continuous on|z| ≤1 except maybe atz = 1. Iff(eı θ)≤Afor 0 < θ <2π, then
|f(z)| ≤A for all |z| <1.
6.33. Let D denote the unit disk {z ∈ C;|z| < 1}, and let {fn} be a sequence of holomorphic functions in D such that lim
n→∞fn = f uniformly on compact subsets of D.
Suppose that each fn takes on the value 0 at most seven times on D (counted with multiplicity). Prove that either f ≡ 0 or f takes on the value 0 at most seven times on D (counted with multiplicity).
6.34. Show that the function f(z) = z + 2z2 + 3z3 + 4z4+· · · is injective in the unit disk D={z ∈C;|z|<1}. Find f(D).
6.35. Suppose f is a nonconstant function holomorphic on {z ∈ C;|z|<1}and continuous on {z ∈C;|z| ≤1} such that for all θ ∈R, the value f(eıθ) is on the boundary of the triangle with vertices 0, 1, and ı.
EXERCISES 121
Is there a z0 with |z0| < 1 such that f(z0) = 101(1 +ı)? Is there a z0 with|z0|<1 such that f(z0) = 12(1 +ı)?
6.36. Is there a function f holomorphic for |z|<1 and continuous for |z| ≤1 that satisfies
f(eı θ) = cosθ+ 2ı sinθ, for all θ∈R?