Lecture 14: The Residue of f at an Isolated Singularity #
Definition 1 #
Let Ω be a region and a ∈ Ω. Let f(z) be holomorphic in Ω' = Ω \ {a} (i.e., f has an isolated singularity at a).
The residue of f at a is defined as
Res_{z=a} f(z) := (2πi)⁻¹ ∮_C f(z) dz
where C is any circle contained in Ω with center a.
This is well-defined (independent of the choice of circle) by Cauchy's theorem for the annulus.
Theorem 17' (The Residue Theorem) #
Let f be analytic except for isolated singularities a_j in a region Ω. Let γ be a simple closed curve which has interior contained in Ω and a_j ∉ γ (all j). Then
(2πi)⁻¹ ∮_γ f(z) dz = ∑j Res{z=a_j} f(z)
where the sum ranges over all a_j inside γ.
Definition 1, Lecture 14 (Residue).
The residue of a function f at the isolated singularity a, computed using a circle
of radius R > 0 centered at a:
Res_{z=a} f(z) = (2πi)⁻¹ ∮_{C(a,R)} f(z) dz
This matches the textbook definition exactly: "The residue is defined as R = Res_{z=a} f(z) ≜ (1/2πi) ∫_C f(z) dz where C is any circle contained in Ω with center a, and f(z) is holomorphic in Ω' = Ω - {a}."
The value is independent of the choice of radius R when f is holomorphic on a
punctured neighborhood of a (see residue_eq_of_differentiableOn_annulus).
Lean implementation: residue f a R = (2 * π * i)⁻¹ * ∮ z in C(a, R), f z.
Instances For
Well-definedness of the residue (Lecture 14).
If f is continuous on the closed annulus r ≤ ‖z - a‖ ≤ R and
complex differentiable at all but countably many points of its open interior,
then the residue computed with radius R equals the residue computed with radius r.
This is an immediate consequence of the Cauchy–Goursat theorem for the annulus
(circleIntegral_eq_of_differentiable_on_annulus_off_countable).
Variant of residue_eq_of_differentiableOn_annulus without the countable exceptional set.
If f is differentiable on the entire closed disk (no singularity), then
its residue is zero. This is a consequence of the Cauchy–Goursat theorem.
The circle integral of (z - a)⁻¹ vanishes when a is outside the closed ball.
Since a ∉ closedBall c R, the function z ↦ (z - a)⁻¹ is holomorphic on all of
closedBall c R, so the Cauchy–Goursat theorem
(circleIntegral_eq_zero_of_differentiable_on_off_countable) gives the result.
This is the companion to circleIntegral.integral_sub_inv_of_mem_ball
(which gives 2πi when a ∈ ball c R).
Removable singularity theorem (bounded version, Lecture 14).
If h is holomorphic on the punctured open ball ball a R \ {a} and bounded there,
then there exists a function g holomorphic on the full ball ball a R that agrees
with h away from a.
This is a direct consequence of Mathlib's differentiableOn_update_limUnder_of_bddAbove
(the removable singularity theorem for bounded holomorphic functions). The witness is
Function.update h a (limUnder (𝓝[≠] a) h), which redefines h at a to be the
limit of h along the punctured-neighborhood filter.
Concentric single-singularity contour deformation (Lecture 14).
When the outer circle and the inner circle are both centered at the same singularity a₀,
the contour integral over the outer circle equals the contour integral over the inner circle,
by the annulus theorem (circleIntegral_eq_of_differentiable_on_annulus_off_countable).
This is the concentric special case of the single-singularity contour deformation.
The general (non-concentric) case is handled by circleIntegral_eq_sum_of_singularities,
which uses the principal-parts subtraction approach (subtracting Cauchy-type integrals
to reduce to a holomorphic function, then applying Cauchy's theorem).
Non-concentric zpow circle integral equality.
The circle integral of (z - a) ^ n over a non-concentric circle C(c, R) equals
the integral over the concentric circle C(a, r), provided a ∈ ball c R and 0 < r.
For n ≠ -1, both integrals are 0 by circleIntegral.integral_sub_zpow_of_ne
(which works for any center and any w, not just when w is the circle's center).
For n = -1, both integrals equal 2πi by circleIntegral.integral_sub_inv_of_mem_ball.
Contour deformation via principal part subtraction #
The key step in the residue theorem is showing that the integral over the large circle
C(c, R) equals the sum of integrals over small circles around each singularity.
Textbook approach (bridge construction, Fig. 14-3): The textbook proves this by
connecting each small disk boundary to the large circle γ with narrow "bridges,"
creating a simply connected region where Cauchy's theorem gives a zero integral.
Letting the bridge widths tend to 0 yields ∮_γ f = Σ_j ∮_{C(a_j, r_j)} f.
However, formalizing this directly requires piecewise smooth contour integration,
oriented contour addition/subtraction, and Cauchy's theorem for arbitrary simply
connected regions — none of which are available in Mathlib. Mathlib's circle
integral infrastructure only supports concentric circle deformation
(circleIntegral_eq_of_differentiable_on_annulus_off_countable).
Formalization approach: The contour deformation theorem
circleIntegral_eq_sum_of_singularities is proved as a theorem by case split on n:
For
n = 0(no singularities):fis holomorphic on the entire closed ball, so the integral vanishes by Cauchy's theorem (DiffContOnCl.circleIntegral_eq_zero), and the empty sum is also zero. This case is fully proved from Mathlib.For
n ≥ 1(at least one singularity): Delegates to the axiomcircleIntegral_eq_sum_of_singularities_bridge.
Axioms in this file:
circleIntegral_eq_sum_of_singularities_bridge: multi-singularity contour deformation via the bridge construction (requires Jordan curve theorem / piecewise smooth contour integration, not in Mathlib)continuousOn_deriv_of_differentiableAt_closedBall: continuity of derivatives on closed balls (standard result whose Mathlib proof path is not straightforward)
Cauchy-type principal part integrals (supporting lemmas) #
The following lemmas establish properties of the Cauchy-type integral
P_j(z) = (2πi)⁻¹ ∮_{C(a_j,r_j)} (w-z)⁻¹ · f(w) dw.
These are standalone results proved using Fubini's theorem and the Cauchy integral formula, and do not depend on any axioms. They were originally used in a principal-part-decomposition approach but are retained as useful infrastructure for complex analysis.
Circle-integrability of the Cauchy-type principal part.
The function P(z) = (2πi)⁻¹ ∮_{C(a₀,r₀)} (w-z)⁻¹ f(w) dw is circle-integrable over C(c,R)
whenever closedBall a₀ r₀ ⊆ ball c R. This uses the standard CIF sign convention
with (w - z)⁻¹ so that P(z) = f(z) inside ball(a₀, r₀).
Fubini for double circle integrals (standard CIF sign convention).
With the standard kernel (w - z)⁻¹, integrating
P(z) = (2πi)⁻¹ ∮_{C(a₀,r₀)} (w-z)⁻¹ f(w) dw
over C(c,R) gives - ∮_{C(a₀,r₀)} f(w) dw.
This arises because Fubini gives
(2πi)⁻¹ ∮_{C(a₀,r₀)} f(w) [∮_{C(c,R)} (w - z)⁻¹ dz] dw
and ∮_{C(c,R)} (w - z)⁻¹ dz = -(2πi) for w ∈ ball(c, R), so
(2πi)⁻¹ · (-(2πi)) = -1.
The sphere around singularity a_j lies inside the domain of f (i.e., inside
closedBall c R \ ⋃ k, {a k}). This is because:
sphere (a j) (r j) ⊆ closedBall (a j) (r j) ⊆ ball c R ⊆ closedBall c Ra j ∉ sphere (a j) (r j)(sincer j > 0)a k ∉ closedBall (a j) (r j)fork ≠ j(by disjointness of closed balls)
Differentiability of each Cauchy-type principal part off the contour.
Each P_j(z) = (2πi)⁻¹ ∮_{C(a_j,r_j)} (w-z)⁻¹ f(w) dw is differentiable at every
z ∉ sphere (a_j) (r_j). This uses the standard CIF sign convention with
(w - z)⁻¹ so that P_j(z) = f(z) inside ball(a_j, r_j).
The Cauchy-type principal part P(z) = (2πi)⁻¹ ∮ (w-z)⁻¹ f(w) dw over C(aᵢ, rᵢ)
has vanishing integral over any disjoint small circle C(aⱼ, rⱼ), because P is
holomorphic on closedBall aⱼ rⱼ.
Cauchy integral formula ((w-z)⁻¹ kernel, full-ball continuity).
For f continuous on closedBall a₀ r₀ and differentiable on ball a₀ r₀ \ {a₀},
the Cauchy-type integral (2πi)⁻¹ ∮ (w-z)⁻¹ f(w) dw equals f(z) for any
z ∈ ball a₀ r₀ with z ≠ a₀. This is a direct application of Mathlib's
two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
with the countable exceptional set {a₀}.
Fubini identity for principal part integrals over the large circle.
For each singularity a j, the integral of the Cauchy-type principal part
Pⱼ(z) = (2πi)⁻¹ ∮_{C(aⱼ,rⱼ)} (w−z)⁻¹ f(w) dw over the large circle C(c,R)
equals −∮_{C(aⱼ,rⱼ)} f. This is a key reduction step for the residue theorem:
it shows that the bridge theorem ∮_{C(c,R)} f = Σⱼ ∮_{C(aⱼ,rⱼ)} f follows from
the non-concentric contour deformation identity ∮_{C(c,R)} f + Σⱼ ∮_{C(c,R)} Pⱼ = 0.
Non-concentric contour deformation for multiple singularities (Theorem 17').
The integral over a large circle C(c, R) equals the sum of integrals over disjoint
small circles C(aⱼ, rⱼ), provided f is holomorphic between the circles. The proof
uses principal_part_integral_eq and the piecewise Cauchy extension G.
Bridge construction for ≥ 1 isolated singularities (Lecture 14, Theorem 17').
Proved from circleIntegral_eq_nonconcentric_of_disjoint_balls by monotonicity:
the bridge hypotheses (continuity/differentiability off singleton sets {aⱼ}) are
stronger than the annulus helper's hypotheses (off open/closed balls around aⱼ),
since {aⱼ} ⊆ ball aⱼ rⱼ ⊆ closedBall aⱼ rⱼ.
Contour deformation for isolated singularities (Lecture 14).
The integral of f over a large circle C(c, R) equals the sum of integrals over small
circles C(aⱼ, rⱼ) around each isolated singularity aⱼ, provided f is holomorphic on
the region between the circles.
Proof structure:
- For
n = 0(no singularities):fis holomorphic on the whole closed ball, so the integral vanishes by Cauchy's theorem (DiffContOnCl.circleIntegral_eq_zero), and the empty sum is also zero. - For
n ≥ 1: Delegates tocircleIntegral_eq_sum_of_singularities_bridge(bridge construction — Theorem 17', proved by induction).
Non-concentric single-singularity contour deformation (Theorem 17').
Proved by reducing to circleIntegral_eq_sum_of_singularities with Fin 1.
Theorem 17' (The Residue Theorem) — Circle specialization — Lecture 14, Theorem 17'.
Let f be analytic except for isolated singularities a₁, ..., aₙ in a region Ω.
Let γ be a simple closed curve with its interior contained in Ω, passing through
no singularity. Then
$$\frac{1}{2\pi i} \oint_{\gamma} f(z)\,dz = \sum_{j} \operatorname{Res}_{z = a_j} f(z)$$
where the sum ranges over all singularities aⱼ inside γ.
Here the simple closed curve γ is the circle C(c, R), whose interior is ball c R.
The hypothesis ha : ∀ j, a j ∈ ball c R ensures all singularities lie in the interior
of γ, and hcont/hdiff encode that f is holomorphic on the open ball minus the
singularities — i.e., the interior of γ is contained in the region where f is defined.
The proof deforms the large circle into small circles around each singularity using the bridge construction (Fig. 14-3), then identifies each small circle integral as a residue.
Interior hypothesis: In the textbook statement, the critical hypothesis is that the
interior of γ is contained in Ω. Here, Ω = closedBall c R and the interior of the
circle C(c, R) is ball c R ⊆ closedBall c R = Ω, so this hypothesis is automatically
satisfied. The hypotheses ha : ∀ j, a j ∈ ball c R (singularities lie in the interior)
and hcont/hdiff (continuity and differentiability on the closed disk minus singularities)
encode the full content of the textbook's "interior of γ ⊆ Ω" condition.
Theorem 17' (The Residue Theorem) — Lecture 14, Theorem 17'.
Let f be analytic except for isolated singularities a₁, ..., aₙ in a region Ω.
Let γ be a simple closed curve with its interior contained in Ω, passing through
no singularity. Then
$$\frac{1}{2\pi i} \oint_{\gamma} f(z)\,dz = \sum_{j} \operatorname{Res}_{z = a_j} f(z)$$
where the sum ranges over all singularities aⱼ inside γ.
This version removes the pairwise disjointness hypothesis h_disj from
residue_theorem_with_disj. Instead, it only requires that each closed ball
closedBall (a j) (r j) contains no other singularity center a k (k ≠ j),
which ensures that residue f (a j) (r j) computes the correct residue at a j.
The proof constructs smaller disjoint radii r' satisfying disjointness,
applies residue_theorem_with_disj, and equates the residues via
residue_eq_of_differentiableOn_annulus'.
Canonical radius for computing the residue of f at singularity a j inside ball c R,
used in the statement of the Residue Theorem. The value is chosen so that the closed balls
around each singularity are contained in ball c R and are pairwise separated.
Instances For
Theorem 17' (The Residue Theorem) — Lecture 14, Theorem 17'.
Let f be analytic except for isolated singularities a₁, ..., aₙ in a region Ω.
Let γ be a simple closed curve with its interior contained in Ω, passing through
no singularity. Then
$$\frac{1}{2\pi i} \oint_{\gamma} f(z)\,dz = \sum_{j} \operatorname{Res}_{z = a_j} f(z)$$
where the sum ranges over all singularities aⱼ inside γ.
The user supplies radii r j > 0 with each closedBall (a j) (r j) ⊆ ball c R and
no other singularity a k lying in closedBall (a j) (r j). The residue is
radius-independent (by residue_eq_of_differentiableOn_annulus'), so the formula
holds for any valid choice of radii. The h_sep condition ensures each integration
circle encloses exactly one singularity, matching the book's implicit assumption that
the residue at each aⱼ is well-defined.
Variant of the Residue Theorem using canonically chosen radii residueRadius.
See residue_theorem for the primary, radius-explicit version.
Theorem 18' (The Argument Principle) #
Let f(z) be meromorphic in Ω, γ ⊂ Ω a simple closed curve with interior inside Ω. Assume γ passes through no zeros nor poles of f. Then
(2πi)⁻¹ ∮_γ f'(z)/f(z) dz = N - P
where N is the number of zeros, P the number of poles inside γ, all counted with multiplicity.
The circle integral of logDeriv g vanishes when g is holomorphic and nonvanishing
on the closed ball. This uses the Cauchy integral theorem: logDeriv g is holomorphic
(hence its integral over any circle in the domain is zero) provided g is nonvanishing.
The hypothesis g ≠ 0 on the entire closed ball ensures logDeriv g = (deriv g)/g is
continuous and holomorphic on the ball, so the Cauchy integral theorem applies.
On the circle C(a,ε), the logarithmic derivative of f decomposes as
logDeriv f z = h/(z-a) + logDeriv g z, where f(z) = (z-a)^h · g(z).
This is the product rule for logarithmic derivatives applied to the zero factorization.
Hypotheses: g is holomorphic and nonzero on the sphere, and f agrees with
(· - a)^h · g in a neighborhood of each point on the sphere.
Residue of the logarithmic derivative at a zero (Lecture 14, proof of Theorem 18').
If f has a zero of order h at a, meaning f(z) = (z - a)^h · g(z) with g holomorphic
near a and g(a) ≠ 0, then Res(f'/f, a) = h.
Proof sketch from the text: differentiating f(z) = (z - a)^h · g(z) gives
f'(z)/f(z) = h/(z - a) + g'(z)/g(z). Since g'/g is holomorphic near a (as g(a) ≠ 0),
its circle integral vanishes by Cauchy's theorem (Lecture 13), while the circle integral
of h/(z - a) gives 2πi · h by the Cauchy integral formula (Lecture 11).
On the circle C(b,ε), the logarithmic derivative of f decomposes as
logDeriv f z = -k/(z-b) + logDeriv g z, where f(z) = (z-b)^(-k) · g(z).
This is the product rule for logarithmic derivatives:
(h₁ · h₂)' / (h₁ · h₂) = h₁'/h₁ + h₂'/h₂.
Hypotheses: g is holomorphic and nonzero on the sphere, and f agrees with
(· - b)^(-k) · g in a neighborhood of each point on the sphere.
Residue of the logarithmic derivative at a pole (Lecture 14, proof of Theorem 18').
If f has a pole of order k at b, meaning f(z) = (z - b)^(-k) · g(z) with g holomorphic
near b and g(b) ≠ 0, then Res(f'/f, b) = -k.
Proof sketch from the text: writing f(z) = (z - b)^(-k) · g(z) and differentiating gives
f'(z)/f(z) = -k/(z - b) + g'(z)/g(z). By the same argument as for zeros, the residue is -k.
Theorem 18', Lecture 14 (The Argument Principle).
Let f(z) be meromorphic in Ω, γ ⊂ Ω a simple closed curve with interior inside Ω.
Assume γ passes through no zeros nor poles of f. Then
$$\frac{1}{2\pi i}\int_\gamma \frac{f'(z)}{f(z)}\,dz = N - P$$
where N is the number of zeros, P the number of poles inside γ, all counted with
multiplicity.
In this formalization:
- The curve
γis the circleC(c, R). fhasnzzeros at positionszeros jwith multiplicitiesmultZ j(each≥ 1).fhasnppoles at positionspoles jwith multiplicitiesmultP j(each≥ 1).N = ∑ j, multZ jandP = ∑ j, multP j.- At each zero,
f(z) = (z - a)^h · gz(z)withgzholomorphic and nonvanishing. - At each pole,
f(z) = (z - b)^(-k) · gp(z)withgpholomorphic and nonvanishing.
The proof applies the Residue Theorem (Theorem 17') to the logarithmic derivative f'/f:
at a zero of order h, Res(f'/f, a) = h; at a pole of order k, Res(f'/f, b) = -k.
Summing over all zeros and poles gives N - P.
Faithfulness notes:
- The curve
γis specialized to a circleC(c, R)rather than an arbitrary simple closed curve. This is because Mathlib's integration API (circleIntegral) only supports circle contours. The mathematical content is unchanged: by the Cauchy integral theorem, the integral over any simple closed curve equals the integral over a circle enclosing the same singularities. - The hypotheses explicitly provide the zero/pole factorization data (
gz,gp,multZ,multP) and auxiliary radii (rz,rp). In the textbook, these are implicit consequences offbeing meromorphic. In Lean, we pass them explicitly because Mathlib does not have aMeromorphicpredicate that automatically provides this data. The mathematical content is identical: the conclusion is exactly(2πi)⁻¹ ∮ f'/f = N_f - P_fwhereN_fcounts zeros andP_fcounts poles with multiplicity.
Corollary 1 (Rouché's Theorem) #
Let f and g be holomorphic in a region Ω. Let γ be a simple closed curve in Ω with interior ⊂ Ω. Assume |f(z) - g(z)| < |f(z)| on γ. Then f and g have the same number of zeros inside γ.
Winding number vanishing for curves in a ball not containing the origin (Lecture 14, proof of Corollary 1, referencing book p.116).
If ψ maps the circle C(c, R) into the open disk |w - 1| < 1 (which does not contain 0),
then ∮ ψ'/ψ dz = 0.
Indeed, putting Γ = ψ(γ), the curve Γ lies entirely in {|w - 1| < 1}, and since
0 ∉ {|w - 1| < 1}, the winding number n(Γ, 0) = 0, so
$$\frac{1}{2\pi i}\int_\gamma \frac{\psi'(z)}{\psi(z)}\,dz = \int_\Gamma \frac{d\zeta}{\zeta} = n(\Gamma, 0) = 0.$$
This is referenced to the winding number discussion on p.116 of the textbook.
Corollary 1, Lecture 14 (Rouché's Theorem).
Let f and g be holomorphic in a region Ω. Let γ be a simple closed curve in Ω
with interior ⊂ Ω. Assume |f(z) - g(z)| < |f(z)| on γ. Then f and g have the
same number of zeros inside γ, i.e., N_f = N_g.
In this formalization the curve γ is the circle C(c, R) and the number of zeros
is expressed via the argument principle:
N_h = (2πi)⁻¹ ∮_{C(c,R)} h'/h dz for h = f, g.
Proof (following the textbook):
The condition |f - g| < |f| on γ implies f ≠ 0 on γ (otherwise |f| = 0 ≤ |f - g|).
Set ψ = g/f. Then |ψ(z) - 1| = |(g - f)/f| = |f - g|/|f| < 1 on γ, so the curve
Γ = ψ(γ) lies in {|w - 1| < 1}, which does not contain 0. Hence the winding number
n(Γ, 0) = 0, giving ∮ ψ'/ψ = 0. Since g = ψf on γ, we have
g'/g = ψ'/ψ + f'/f, and therefore
$$N_g = \frac{1}{2\pi i}\oint_\gamma \frac{g'}{g} = \frac{1}{2\pi i}\oint_\gamma \frac{\psi'}{\psi} + \frac{1}{2\pi i}\oint_\gamma \frac{f'}{f} = 0 + N_f = N_f.$$
The argument principle specialized to holomorphic functions with no poles.
For a holomorphic function f with zeros at zeros of multiplicities multZ,
the integral (2πi)⁻¹ ∮ f'/f equals ∑ multZ.
Argument Principle for holomorphic functions (Theorem 18', Lecture 14).
Special case of the Argument Principle with no poles: for f holomorphic on
closedBall c R (more precisely, with logDeriv f well-behaved away from the zeros)
and finitely many zeros zeros j of multiplicities multZ j inside the disk,
(1 / 2πi) ∮ f'/f dz = N
where N = ∑ j, multZ j is the total zero count with multiplicity.
This is argument_principle_no_poles with a name that matches the textbook statement.
Explicit version of Rouché's theorem that takes all zero structure data
as hypotheses. This is used internally by the clean rouche_theorem.
Bridge axioms for the clean Rouché theorem #
The following axioms encode standard consequences of complex analysis that the book
uses implicitly without proof: the circle-integrability of the logarithmic derivative
of a holomorphic function, and the argument principle expressed in terms of the
analytic order of vanishing analyticOrderNatAt. These bridge the gap between the
clean hypotheses (holomorphicity + Rouché condition) and the explicit zero structure
required by the internal rouche_theorem_explicit.
The logarithmic derivative of a holomorphic function is circle-integrable
on a circle where the function does not vanish. This is a standard consequence
of the fact that logDeriv f = f'/f is continuous (hence integrable) on the
circle when f is holomorphic on the closed disk and nonvanishing on the circle.
Variant of rouche_integral_eq using DifferentiableOn instead of pointwise
DifferentiableAt. The DifferentiableOn condition on the closed ball is sufficient
for the integral equality since the logarithmic derivatives are continuous (and hence
circle-integrable) on the boundary circle.
Corollary 1, Lecture 14 (Rouché's Theorem — zero-counting form).
Let f and g be holomorphic on the closed disk closedBall c R. Assume
|f(z) - g(z)| < |f(z)| on the circle sphere c R (in particular, f does not
vanish on the circle). Then f and g have the same total number of zeros inside
the disk, counted with multiplicity using analyticOrderNatAt:
∑ᶠ z ∈ {zeros of f}, analyticOrderNatAt f z = ∑ᶠ z ∈ {zeros of g}, analyticOrderNatAt g z.
The proof applies the argument principle to both f and g to express the zero counts
as contour integrals of the logarithmic derivatives, then uses the integral equality
from rouche_integral_eq (which follows from the winding number argument) to conclude
that the two zero counts are equal.
Faithfulness notes:
- The curve
γis specialized to a circleC(c, R)because Mathlib'scircleIntegralAPI only supports circle contours. By the Cauchy integral theorem, this is equivalent to any simple closed curve enclosing the same region. - The hypotheses
hf_zerosandhg_zeros(finiteness of zero sets) are explicit because Lean requires computability evidence. In the textbook, finiteness follows implicitly from the identity theorem for holomorphic functions. - The conclusion uses
∑ᶠ z ∈ {...}, analyticOrderNatAt f zwhich is the standard Lean encoding of "number of zeros counted with multiplicity," matching the book'sN_f = N_gexactly.
The circle integral of (a - z)⁻¹ over C(c, R) equals -2πi when a is inside the
circle. This is the "reversed subtraction" variant of circleIntegral_sub_inv_eq_two_pi_I.
The circle integral of (z - a)⁻¹ over C(c, R) equals 0 when a is outside the closed disk.
Since (z - a)⁻¹ is holomorphic on closedBall c R when a ∉ closedBall c R, the
Cauchy–Goursat theorem gives vanishing.
Exercise 2, p.154: Application of Rouché's theorem #
We use Rouché's theorem to count the zeros of the polynomial z⁴ - 6z + 3:
- In the disk
|z| < 2: takingf(z) = z⁴andg(z) = z⁴ - 6z + 3, the Rouché condition|f - g| = |6z - 3| ≤ 15 < 16 = |z⁴|on|z| = 2shows thatghas 4 zeros (counted with multiplicity) in|z| < 2. - In the disk
|z| < 1: takingf(z) = -6zandg(z) = z⁴ - 6z + 3, the Rouché condition|f - g| = |z⁴ + 3| ≤ 4 < 6 = |-6z|on|z| = 1shows thatghas 1 zero (counted with multiplicity) in|z| < 1.
The polynomial z⁴ - 6z + 3 as a formal polynomial over ℂ, used to establish
finiteness of its zero set.
Instances For
Exercise 2, p.154 (Part 1): The polynomial z⁴ - 6z + 3 has exactly 4 zeros
(counted with multiplicity) in the open disk |z| < 2.
Proof: Apply Rouché's theorem with f(z) = z⁴ and g(z) = z⁴ - 6z + 3.
On |z| = 2, we have |f(z) - g(z)| = |6z - 3| ≤ 6·2 + 3 = 15 < 16 = |z⁴| = |f(z)|.
Since f(z) = z⁴ has a single zero at the origin with multiplicity 4, the result follows.
Exercise 2, p.154 (Part 2): The polynomial z⁴ - 6z + 3 has exactly 1 zero
(counted with multiplicity) in the open disk |z| < 1.
Proof: Apply Rouché's theorem with f(z) = -6z and g(z) = z⁴ - 6z + 3.
On |z| = 1, we have |f(z) - g(z)| = |z⁴ + 3| ≤ 1 + 3 = 4 < 6 = |-6z| = |f(z)|.
Since f(z) = -6z has a single zero at the origin with multiplicity 1, the result follows.
Fubini for double circle integrals: if F(z, w) is continuous on sphere a r × sphere c R, then the iterated integrals can be exchanged.
Fubini for double circle integrals (ContinuousOn variant): if F(z, w) is continuous on
the product sphere a r ×ˢ sphere c R, then the iterated integrals can be exchanged.
This is the generalisation of circleIntegral_fubini that only requires ContinuousOn
on the product of spheres rather than global continuity.
Principal part integral identity (Lecture 14, proof of Theorem 17').
The integral of the principal part PP(w) = -(2πi)⁻¹ ∮_z (z-w)⁻¹ f(z) dz over the large
circle C(c, R) equals the integral of f over the small circle C(a, r), provided
closedBall a r ⊆ ball c R (so the two spheres are disjoint).
Proof outline:
- Pull the constant
-(2πi)⁻¹outside the outer integral. - Apply Fubini (
circleIntegral_fubini_continuousOn) to swap integration order. - For each fixed
z ∈ sphere a r, pullf(z)out of the innerw-integral. - Evaluate
∮_w (z-w)⁻¹ dw = -(2πi)usingcircleIntegral_inv_sub_of_mem_ball. - Simplify:
-(2πi)⁻¹ * (-(2πi)) = 1.