Ph. DQualifyingExamination j - Zhejiang University · Ph. DQualifyingExamination ... j C 1exp(C...

Ph. D Qualifying ExaminationComplex analysisFall 1998

Work all problems. All problems have equal weight. Write the solution to each problemin a separate bluebook.

1. Let f(z) be an entire function that is not a polynomial. Assume f has only finitelymany zeros. Let m(r) = min|z|=r |f(z)|. Show m(r)→ 0 as r →∞.

2(a). Let f(z) be analytic in a bounded region Ω and be continuous up to the boundaryB of Ω. Let E = f(B). If a and b are in the same component of C \E, show that a and bare taken the same number of times by f .

(b). Prove that z4 + z2 + 2 has a zero in each quadrant.

3. Show:∫ ∞

0

xα

(x+ 2)2dx =

απ2α−1

sinπαfor −1 < α < 1.

4. Show that the annulus can be mapped conformally onto the Riemann sphere with twosegments of the real axis removed.

Hint: You may use the Riemann mapping theorem, including boundary behavior.

5(a). Let f(z) be an entire function such that |f(z)| < A exp |z|α. Show that the numberof zeros of f in the disk |z| < r is ≤ Crα for some constant C and for r > 1.

(b). Let f(z) =∑

znn−an. Show that f satisfies

|f(z)| ≤ C1 exp(C2|z|1/α) for some C1, C2 > 0.

6. Let Ω be a bounded connected region. Show that there is an analytic function f(z)defined in Ω such that f cannot be extended to be analytic in any larger connected region.

1

Ph. D Qualifying ExaminationComplex analysisSpring 1998


1. Evaluate∫ ∞

−∞

sin2 x

x2dx

by contour integration.

2(a). Prove that

f(z) =∑

2−n2

zn

has infinitely many zeros.

(b). Suppose f is entire, |f(z)| ≤ eA|z|, and f(z) = f(−z). Prove that either f is apolynomial or f has infinitely many zeros.

3(a). Give an example of a region D in C whose complement is an infinite set and suchthat there is no conformal map of D onto a bounded region.

(b). Let Ω be the complement of eiθ : −π2 ≤ θ ≤π2 with the point at infinity adjoined.

Find a function f(z) that maps Ω conformally onto the unit disk.

4(a). Let |f(z)| assume its maximum on D = z : |z| ≤ 1 at z0, where f is analytic in aneighborhood of D. Show that f ′(z0) 6= 0.(b). Assume f is entire, f(z + 1) = f(z), and |f(z)| ≤ ec|z| for some c < 2π. Show thatf is constant.

5. Assume that f is analytic in a neighborhood of D = z : |z| ≤ 1. Assume also that|f(eiθ)| ≤ m if −π2 ≤ θ ≤ π

2 and that |f(eiθ)| ≤ M if π2 ≤ θ ≤ 3π2 . Find the best possible

bound for |f(0)|.

Hint: First assume that f has no zeros.

6. Let u(z) be a real harmonic function on z : 1 < |z| < 2. Show that for some α andfor some function f(z) analytic in z : 1 < |z| < 2,

u(z) = α log r +Re f(z).

1


Work all six problems.

1. Show that for a > 0,∫ ∞

0

cos ax

(1 + x2)2dx =

π(a+ 1)e−a

4if a > 0.

2. Let D be the open unit disk centered at the origin and let f : D → C be a function.Suppose f is analytic in D, f is continuous in D \ 1 and

limz→1

|f(z)|log |z − 1|

= 0.

Suppose further that |f(w)| ≤ 1 for w ∈ ∂D \ 1. Show that

maxz∈D

(|f(z)|) ≤ 1.

3. Let f(z) be an analytic function on the punctured disk 0 < |z| < 1 with 0 an essentialsingularity. Let ξ be any complex number. Show that, with the possible exception of onevalue of ξ,

limr→0+

1

2π√−1

∫

|z|=r

f ′(z)

f(z)− ξdz =∞,

where the limit is taken for those r > 0 for which f(z)− ξ has no zeros on |z| = r.

4(a). Let Ω be the (open) ball of radius 1 centered at the origin and E be a compactsubset in Ω. Use Poisson’s formula for harmonic functions to prove the following versionof the Harnack inequality: There is a constant M , depending only on E, such that everypositive harmonic function u(z) in Ω satisfies

u(z1) ≤Mu(z2), z1, z2 ∈ E.

(Do not quote Harnack’s inequality directly.)

4(b). Find the best possible M in case E is the closed disk of radius 12 centered at the

origin.

5. Let D be the interior of the triangle whose vertices are 0, 1 and i.

(a) Prove that there is a unique conformal mapping w = f(z) of D onto the upper halfplane such that

limz→0

f(z) = 0, limz→1

f(z) = 1 and limz→i

f(z) =∞.

1

(b) Prove that f extends by reflection to an elliptic function. Find the poles and theperiods of this function.

(c) Find the singular part of this function near its poles. Find an explicit formula forf as an infinite sum.

(Hint. For (c), study the rotational symmetry of f near a pole.)

6. Let H be the upper half-plane and let F (z) be the function defined by

F (z) =

∫ z

0

dw

(1− w)(1 + w)√w, z ∈ H

where the integral is over a path from 0 to w in H and where√w is the single-value branch

of the square root function such that√1 = 1. Show that F is a one-one conformal map

onto the region

Ω = (x, y) : x > 0, y > 0,min(x, y) <π

2.

2


Work all 6 problems. All problems have equal weight. Write each solution in a separatebluebook.

1. Let f(x) : (− 12 ,∞)→ C be a continuous function. Suppose f is analytic in a neighbor-

hood of the origin and that there is a positive constant N so that

limx→∞

f(x)eNx = 0.

For the complex variable s we define

F (s) =

∫ ∞

0f(x)xsdx.

(a) Show that the integral converges for Re (s) > −1 and that F (s) has a meromorphiccontinuation to all s with possible poles only at s = −1,−2, · · · .

(b) Determine the exact location of the poles of F and its singular parts at poles.

Hint: The answer to the second part depends on the coefficients of the Taylor expansionof f at x = 0.

2. Let D be a bounded region in C whose boundary consists of n-smooth disjoint Jordanarcs. Thus D is n-connected. We denote by D the closure of D.

(a) Suppose f(z) is a non-constant continuous function on D and is analytic in D.Suppose further that

(*) |f(w)| = 1 for allw ∈ ∂D.

Show that f has at least n zeros (counting multiplicities) in D.(b) For any n > 0, find an n-connected region D and an analytic function f :D → C

such that f satisfies (*) and has exactly n zeros in D.

Remark: You will receive partial credits if you work out the special case where D = 1 <|z| < c in part (a) and/or (b).

3. Show that∫ ∞

0

x−c

1 + xdx =

π

sinπcif 0 < c < 1.

Remark: You need to provide details to justify each step in your computation.

4. Let D be the open unit disk and f :D → C be a bounded analytic function.

(a) Let ann≥1 be the non-zero zeros of f in D counted according to multiplicity. Weassume an is an infinite sequence. Then prove that

∞∑

n=1

(1− |an|) <∞.

1

(b) Let f and an be as above. We define

B(z) =∞∏

n=1

|an|an

( an − z1− anz

)

.

Show that B(z) is a bounded analytic function on D with zeros an. Show furtherthat there is an integer m and a bounded non-vanishing holomorphic function h(z)so that

f(z) = zmB(z)h(z).

5. Let D be the open unit disk and let f be a non-constant analytic function in D.

(a) Suppose for every a ∈ ∂D \ 1 we have

∗ limz→a|f(z)| ≤ 1

and for any δ > 0 we have

∗∗ limz→1|f(z)||z − 1|δ = 0.

Show that |f(z)| < 1 in D.(b) Construct an analytic function on the unit disk D that is not bounded in D and

that satisfies (∗) but not (∗∗).Hint: Consider u(z) = (z − 1)δf(z).

6. Let ω1 and ω2 be two non-zero complex numbers with non-real ratio ω1/ω2. Let Λ bethe lattice Λ = Zω1+Zω2 and let a and b be two complex numbers not congruent to eachother. We form the linear space V of all elliptic functions of period Λ with at most simplepoles at a and b.

(a) Prove that dimC V is at most 2.(b) Using the method of infinite series, construct explicitly a two dimensional families

of elliptic functions in V , thereby proving that dimV = 2.

Remark: In (b) one needs to provide details to why the series converges, why they haveperiod Λ and why they provide a two dimensional family.

2



1. Letf(z) = zn + a1z

n−1 + · · ·+ an

be a polynomial with complex coefficients a1, . . . , an. Let αk be the real part of ak. Supposef(z) has n zeros in the upper-half plane Im z > 0. Prove that the polynomial

α(x) = xn + α1xn−1 + · · ·+ αn

has n distinct real roots.

2. Let D be the open unit disk and let f : D → C be an odd univalent function. (Thatis, f(−z) = −f(z) and f is one-one.) Show that there is a univalent analytic functiong : D → C such that

f(z) =√

g(z2).

3. Let Ω be the region −1 < Re(z) < 1 and let F be the collection of all analytic functionsf(z) defined on Ω such that f(0) = 0 and |f(z)| < 1 for all z ∈ Ω. Find

supf∈F

∣

∣

∣

∣

f

(

1

2

)∣

∣

∣

∣

.

4. Define

F (z) =

∫ ∞

0xz−1e−x

2

dx

for Re(z) > 0.

(a) Prove that F is an analytic function on the region Re(z) > 0.(b) Prove that F extends to a meromorphic function on the whole complex plane.(c) Find all poles of F and find the singular parts of F at these poles.

5. Calculate the following integral:

∫ ∞

0

cosx− 1

x2dx.

6. Let Ω be a connected open subset of C.(a) Let h(z) be a non-trivial analytic function defined on Ω. Let ann≥1 be all the

(distinct) zeros of h(z) and let cnn≥1 be a sequence of complex numbers. Show

1

that there is an analytic function H(z) defined on Ω such that H(an) = cn for alln.

(b) Let f(z) and g(z) be two analytic functions defined on Ω with no common zeros inΩ. Assume that both f(z) and g(z) have only simple zeros. Prove that there areanalytic functions F (z) and G(z) defined on Ω such that over Ω,

F (z)f(z) +G(z)g(z) = 1.

Hint: One possible approach to (a) is to apply the Mittag-Leffler Theorem for the domainΩ. See below for the exact statement of the theorem. For (b), consider

F (z) =1−G(z)g(z)

f(z).

Mittag-Leffler Theorem: Let bk be a sequence of distinct points in Ω without limitpoints in Ω, and let Pk(z) be a sequence of polynomials without constant terms. Thenthere are meromorphic functions φ defined on Ω such that the poles of φ are the pointsbk and such that (for each k) the singular part of φ at z = bk is Pk(

1z−bk ).

2

Ph.D Qualifying ExaminationComplex analysisSpring 1999-2000

Work all 6 problems. All problems have equal weight. Write each problem in a separatebluebook.

1. a. Let ω1, ω2 be two complex numbers such that ω1/ω2 is not real and let Λ be thelattice n1ω1 + n2ω2|n1, n2 ∈ Z. Prove directly that

ζ(z) =1

z+

∑

ω∈Λ−0

(1

z − ω+

1

ω+

z

ω2)

is a doubly periodic meromorphic function up to constants.b. Prove that the necessary and sufficient condition for the existence of a doubly peri-odic meromorphic function with (finitely many) prescribed poles and singular parts in afundamental parallelogram is that the sum of the residues of the poles is zero.

2. Let ai and bi be real numbers for 1 ≤ i ≤ 3 such that 0 < bi < 1. Let u = u(z) bea holomorphic function in the upper half plane that solves the second order differentialequation

u′′ + u′(b1

z − a1+

b2z − a2

+b3

z − a3) = 0.

a. Solve u and express the solution in integral form.b. Show directly, without quoting the Schwarz-Christoffel theorem, that u maps the realaxis to the boundary of a polygon.c. Determine under what condition on the bi’s does umap the upper half plane conformallyonto a triangle? Prove, again without quoting Schwarz-Christoffel theorem, your assertionand determine the angles of this triangle.

3. Let f be a univalent analytic function defined on the unit disk D centered at the origin.Suppose f(0) = 0. Show that the function

g(z) =√

f(z2)

has a single-valued branch, and is also univalent.

(Hint: Univalent means one-one. Use√

f(z2) = z√

f(z2)z2 .)

4. Let f be a nonconstant harmonic function in the unit disk D which restricts smoothlyto the boundary. Assume f(0) = 0 and ∇f(0) = 0. Show that f(x) = 0 for at least fourdistinct points x on the boundary.

5. Let ∆ be the half-stripRe z > 0, −π < Im z < π

and let L be the boundary of ∆ consisting of three straight pieces. We orient L so that ∆is on the right hand side of L.

1

a. Show that the integral

1

2πi

∫

L

eeζ

ζ − zdζ = E(z)

defines an analytic function on the complement of ∆.b. Show that E(z) extends to an entire function on C.c. Show that E(z) assumes real values for real z.

6. Let f(z) be an entire function that satisfies the relation

pn(z)(f(z))n + pn−1(z)(f(z))

n−1 + · · ·+ p0(z) = 0,

where p0(z), · · · , pn(z), not all trivial, are rational functions on C. Show that f must bea polynomial.

2

Ph.D Qalifying ExaminationComplex Analysis Fall 2001

Do all six problems, each in a separate blue book. All problems have equal weight.

1. Let Ω be the domain |z| < 2 and let [0, 1] ⊂ Ω be the line segment between 0 and 1.

(a) Suppose f : Ω → C is a continuous function that is analytic over Ω \ [0, 1]. Showthat f is analytic over Ω.

(b) Show that there are bounded analytic functions f : Ω \ [0, 1]→ C that can not beextended to analytic functions over Ω.

2. Let f(z) be an analytic function defined over |z| < 1 that maps |z| < 1 one-one andonto C \ (−∞,−1/4]. Find the most general form of f(z).

3. Establish the following identity

( 1

et − 1− 1

t+

1

2

)1

t= 2

∞∑

n=1

1

t2 + 4n2π2.

(Hint: Consider the difference of the terms on the left and the right hand side.)

4. Let f and g0 6= 0, g2, · · · , gn be analytic functions over the punctured disk D∗ = 0 <|z| < 1. Suppose g0, · · · , gn all have at most poles at z = 0. Suppose further that fsatisfies the identity

g0fn + g1f

n−1 + · · ·+ gn = 0.

Show that f has at most a pole at z = 0.

5. Let B be the square |x| ≤ 1, |y| ≤ 1 and let T be the boundary of B, in the xy-plane.Let f : T → R be a piecewise smooth function defined over T . Show that there is asequence of real polynomials pn(x, y) so that pn converges uniformly to f on T .(Hint: The Runge’s approximation theorem might be helpful.)

6. Let f(z) be an analytic function defined over the disk |z| < 2. Show that

∫ 1

0f(x)dx =

1

2πi

∮

|z|=1f(z) log z dz.

Here the path integral is taken counterclockwise.

1


Work all six problems. All problems have equal weight. Write the solution to each problemin a separate bluebook.

1. Let P (z) be a polynomial in z.

(a) Assume that P (z) 6= 0 for Re(z) > 0. Show that P ′(z) 6= 0 for Re(z) > 0. (Hint:Try taking a logarithmic derivative.)

(b) Show that for any polynomial P (z), the zeroes of P ′ are contained in the convexhull of the zeroes of P .

2(a). Let f(z) be an entire function with Re(f(z)) < c(1+|z|)p for some positive constantsc and p. Show that f is a polynomial.(b). Construct an entire function f(z) that is not a polynomial such that for every ε > 0,

limr→∞

e−εrM(r) = 0,

where M(r) = max|z|=r |f(z)|. Justify your answer.

3. Let u(z) be a continuous real-valued function on C that satisfies

u(z) =1

2π

∫ 2π

0u(z + reiθ) dθ

for all z and all r > 0.

(a) Prove that supΩ u = sup∂Ω u for every bounded smooth domain Ω.(b) Show that u is a smooth harmonic function, i.e., that ∆u = 0. (Hint: You may

quote the solution for the Dirichlet problem.)

4(a). Find the radius of convergence of the power series∑∞n=1 nz

n. Find the largestdomain to which the corresponding analytic function can be analytically extended.(b). Suppose that the power series

∑∞n=1 anz

n converges for |z| < 1 and that thecoefficients an are real and nonnegative. Show that if the corresponding analytic functionhas an analytic continuation to a neighborhood of z = 1, then the radius of convergenceof the series is greater than 1.

5. Let u be a subharmonic function defined on C and let M(r) = max|z|=r u(z).

(a) Prove that

u(z) ≤ log r2 − log |z|log r2 − log r1

M(r1) +log |z| − log r1log r2 − log r1

M(r2)

for 0 < r1 ≤ |z| ≤ r2. (Hint: use the maximum principle.)(b) Show that limr→∞M(r)/ log r exists (possibly infinite).(c) Show that if the the limit λ(u) in part (b) is 0, then u is a constant function.

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6(a). Let Γ be a smooth simple closed curve in C. Describe the set of points z such that|P (z)| ≤ maxw∈Γ |P (w)| for all polynomials P (z). Justify your answer.(b). Let L be a line in C and let z0 be a point not on L. Show that there is a sequence

of polynomials Pj(z) with Pj(z0) = 1 such that Pj → 0 uniformly on compact subsets ofL.

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Ph. D Qualifying ExaminationComplex analysisAutumn 2002


1. Let P (w, z) = a0(z)wn + a1(z)w

n−1 + · · · + an(z) be a polynomial in two variables.Suppose z0 ∈ C is a point such that a0(z0) 6= 0 and P (w, z0) has n-distinct zeros, sayw1, · · · , wn. Show that there exists an open disc ∆ ⊂ C containing z0 and n-holomorphicfunctions fi(z) : ∆→ C, i = 1, · · · , n, such that: (1) P (fi(z), z) = 0 on ∆; (2) fi(z0) = wiand (3) whenever P (w, z) = 0 and z ∈ ∆, then w = fi(z) for some i.

2. Find the Green’s function for the region consisting of the complement in the C-planeof the intervals (−∞,−1] and [1,∞) on the real axis.

3. Let H be the upper half plane and H be its closure in C. Let f :H → C be a continuousfunction that is analytic in H. Suppose f is bounded on H and

limt→±∞

f(t) = 0, t ∈ R.

Show thatlim|z|→∞

f(z) = 0, t ∈ H.

(Hint 1: You can use the following result if you like: Let D be the open unit disk andg : D − 1 → C be a continuous map that is analytic in D. Suppose |g| ≤ M on D and|g(eiθ)| ≤ 1 for 0 < θ < 2π. Then |g(z)| ≤ 1 on D.)

(Hint 2: Consider functions of the form log zA+B log z f(z).)

4. Use Argument Principle to show that the function f(z) = eπz−e−πz assumes any valuew with positive real part once and only once in the half strip Re z > 0, − 1

2 < Im z < 12 .

5. Show that

π2cosπz

sin2 πz=

∞∑

n=−∞

(−1)n

(z − n)2.

(Hint: Study the principal parts at the poles and use periodicity.)

6(a). Describe the Riemann surface of w =√

z(z − 1)(z − λ), λ 6= 0, 1.

(b). Show that dz/w is a holomorphic differential on the Riemann surface and describethe mapping defined by

f(ζ) =

∫ ζ

ζ0

dz

w,

where ζ is a general point on the Riemann surface and ζ0 is a chosen basepoint.



Convention: We denote by ∆ the open unit disk and by A the closure of A in C.

1. It is known that for any smooth complex valued function g on ∆, there is a smoothfunction in ∆ so that ∂u

∂z = g. Now let Ω ⊂ C be ∆ with the origin and the line segment

[ 12 , 1] deleted:

Ω = ∆ \ z ∈ R : z = 0 or 12 ≤ z ≤ 1.

Prove that given any function f ∈ C∞(Ω) there is a smooth function u ∈ C∞(Ω) so that∂u∂z = f .

Hint: Use the Runge approximation theorem.

2. (1). Write down an entire function f : C→ C in the form of an infinite product so thatits set of zeros equals logn : n = 2, 3, 4, . . . .(2) State the definition of an entire function being finite order.(3) Is there an entire function of finite order whose zero set is logn : n = 2, 3, 4, . . . ?Prove the existence or the non-existence of such functions.

3. Prove that1

π

∫ ∞

−∞

sin z2 e

i( 12−t)z

zdz =

1 if 0 < t < 1

0 if t < 0 or t > 1.

4. Let ∆∗ be the punctured disc. We let P be the set of all subharmonic functions v on∆∗ such that lim supz→z0 v(z) ≤ 0 for all z0 ∈ ∂∆ and lim supz→0 v(z) ≤ 2002. Prove thatfor any v ∈ P, v(z) ≤ 0 in ∆∗.

5. Show that the function

w = log

(

1 + z

1− z

)

+2z

1 + z2,

maps ∆ one-one and onto the full w-plane with four half-lines deleted. Find the locationsof the four end points of the four half-lines.

6. (1). We let A be the interior of an equilateral triangle. Prove that any one-one, onto

holomorphic map f : A→ ∆ can be extended to a holomorphic function f : U → C definedon an open neighborhood U ⊃ A.(2). We let B be the interior of a triangle whose three interior angles are π

5 ,2π5 and

2π5 . Prove that no one-one, onto holomorphic map f : B → ∆ can be extended to a

holomorphic function f : U → C defined on an open neighborhood U ⊃ B.

Ph. D Qualifying Examination

Complex analysis–Autumn 2003

Work all 6 problems. All problems have equal weight. Write each problemin a separate bluebook.

1. Let D be a bounded region in C whose boundary consists of n-smoothdisjoint Jordan curves. Thus D is n-connected. We denote by D the closureof D.

Suppose f(z) is a non-constant continuous function on D and is analytic inD. Suppose further that

|f(w)| = 1 for all w ∈ ∂D.

Show that f has at least n zeros (counting multiplicities) in D.

2. Show that ∫ ∞

0

(log x)2

(1 + x2)dx =

π3

8.

Hint: You may need to compute∫∞0

1(1+x2)

dx along the way.

Remark: You need to provide details to justify each step in your computa-tion.

3. Let D be the open unit disk and let f : D → C be an odd univalent(i.e. one-one) function. Show that there is a univalent analytic functiong :D → C such that

f(z) =√

g(z2).

1

4. Prove that the function

w = log(z) +z2 − 1z2 + 1

is a 1-1 mapping from the half-plane defined by Re(z) > 0 onto a region Ωin the w plane. Describe the region Ω as explicitly as you can.

5. DefineF (z) =

∫ ∞

0xz−1e−x2

dx.

(a) Prove that F is an analytic function on the region Re(z) > 0.

(b) Prove that F extends to a meromorphic function on the whole complexplane.

(c) Find all the poles of F and find the singular parts of F at these poles.

6. Let ω1 and ω2 be two non-zero complex numbers with non-real ratioω1/ω2. Let Λ be the lattice Λ = Zω1 + Zω2 and let a and b be two complexnumbers not congruent to each other. We form the linear space V of allelliptic functions of period Λ with at most simple poles at a and b.

(a) Prove that dimC V is at most 2.

(b) Using the method of infinite series, construct explicitly a two dimensionalfamilies of elliptic functions in V , thereby proving that dimV = 2.

Remark: In (b) one needs to provide details to why the series converge, whythey have period Λ, and why they provide a two dimensional family.

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PH. D QUALIFYING EXAMINATIONCOMPLEX ANALYSIS–SPRING 2003

1. LetH be the class of analytic functions f(z) on |z| ≤ 1 satisfying f(0) = 0,f ′(0) = 1 and |f(z)| ≤ 100 for all |z| < 1. Prove that there is a constantc > 0 so that for any f ∈ H the image of the unit disk under f contains thedisk |z| < c.

2. Let H be the upper half plane and let F : H → C be defined by

F (z) =∫ z

0

dw

(4− w2)√

w − 1.

Prove, using the argument principle but not quoting the Christoffel-SchwarzLemma directly, that F maps H one-one and onto a domain Ω in C. Identifythis domain Ω.

3. Factor the functioncos

(πz

4)− sin

(πz

4)

into an infinite product.

4. Prove that ∫ π

0ln sin θ dθ = −π ln 2.

5. Let f(z) be an analytic function defined on the unit disk |z| < 1 so thatf(0) = 0 and −1 < Re f(z) < 1 for all |z| < 1. Prove that

|Im f(z)| ≤ 2π

log1 + |z|1− |z| , for0 < |z| < 1.

Find an explicit form of f(z) when equality holds for some 0 < |z| < 1.

6. Let P(z) be the Weierstrass P function of periods 1 and τ . Prove thatthere is a single value branch of the meromorphic function

F (z) =

√P(z)−P(

12)

with F (12) = 0. What are the periods of this function? Verify your assertion.

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