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arXiv:2210.10366v1 [math.CV] 19 Oct 2022

ZEROS OF MEROMORPHIC FUNCTION

LANDE MA AND ZHAOKUN MA

Abstract. We introduce and develop the root locus method in mathematics.

And we study the distribution of zeros of meromorphic functions by root locus

method.

Introduction

The meromorphic functions are the basic functions with general meanings in

mathematics. Meromorphic functions include all complex functions in the engi-

neering. The complex functions that need to be studied in the engineering must

have poles. The poles mainly determine the dynamic characteristics of a process.

Complex functions without poles have no actual values in the engineering. The

main purpose of mathematical research is to provide tools for engineering and other

sciences. In pure mathematics, many functions like the Riemann zeta function and

gamma function, belong to the meromorphic functions. However, there are only

several properties concerning the general meromorphic functions. And, there is

almost no result of its derivative function. Regarding the properties of zeros of

derivative of general meromorphic functions, so far, except for the deﬁnition of the

zero of derivative, nothing has been discovered.

It is well known that the distribution of zeros of the derivative of meromorphic

functions has been studied[8, 9, 10], and several special meromorphic functions are

also studied due to their great inﬂuence, such as Riemann zeta function[11, 12, 13,

14, 15, 16]. By utilizing the results which we obtained, we get the necessary and

suﬃcient results about the zeros of derivatives of meromorphic functions.

Speiser showed that the Riemann hypothesis is equivalent to the absence of non-

trivial zeros of the derivative of the Riemann zeta-function left of the critical line[19].

Levinson and Montgomery proved the quantitative version of the Speiser’s result,

namely, that the Riemann zeta-function and its derivative have approximately the

same number of zeros left of the critical line[12]. We prove there don’t exist the zeros

of derivatives of Riemann zeta function and Dirichlet ate series on the critical line

or on the imaginary axis. We also prove that the results of the values of argument of

Riemann zeta function and Dirichlet ate series strict and monotonic decrease on the

critical line. Riemann hypothesis is equivalent to that argument of xi-function and

Riemann zeta function and Dirichlet ate series concerning the imaginary variable t

strictly and monotonically decrease at the left side of the critical line.

Date: April 19th, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 11M26,11M06 Secondary 30D30,93C05.

Key words and phrases. Root locus; Poles; Zeros; Meromorphic function;

1

2 LANDE MA AND ZHAOKUN MA

1. The results

In textbook of automatic control theory[7], the factor in the left of the root locus

equation is the rational fraction function of the constant coeﬃcient. The root locus

equation is only concerning two degree of 0 and 180 degree which obtains the real

number values. So, the root locus equations and the results of the root locus in

automatic control theory are all very special and limit. The proofs of results in

automatic control theory are not comprehensive and no accurate. For the pure

mathematical study, the factor which is in the left side of the root locus equation

is best substituted by a general meromorphic function. So, we need to give the

results of the root locus of the general meromorphic function.

If the number of poles is ﬁnite or inﬁnite, and the number of zeros is ﬁnite

or inﬁnite, namely the two ﬁnite or inﬁnite products Qm

l=1(1 −s

zl)γlGlz (s) and

Qn

j=1(1 −s

pj)βjGjp (s) both converge, and Kis as: K=|Qn

j=1(1−s

pj)βjGjp(s)

G(s)Qm

l=1(1−s

zl)γlGlz(s)|.

The Kis the reciprocal of the modulus of the meromorphic function G(s)Qm

l=1(1 −

s

zl)γlGlz (s)/Qn

j=1(1 −s

pj)βjGjp (s), let, W(s) = G(s)Qm

l=1(1−s

zl)γlGlz(s)

Qn

j=1(1−s

pj)βjGjp(s)=u(σ, t) +

iv(σ, t), W(s) is a meromorphic function with one complex variable in the extended

complex plane C∪ {∞}. So, after Kand the meromorphic function W(s) are mul-

tiplied, the product result is the unit complex value of the meromorphic function.

Deﬁnition 1.1. In the extended complex plane C∪ {∞}, the equation (1.1) is

called as the root locus equation.

(1.1) KG(s)Qm

l=1(1 −s

zl)γlGlz (s)

Qn

j=1(1 −s

pj)βjGjp (s)=a+ib

In which, a+ib is the unit complex number value of meromorphic function W(s).

The argument of the unit complex number value is as: α= 2qπ+ arg( b

a). The factor

at the left side of the root locus equation (1.1) is meromorphic function W(s).

The zeros zland poles pjare points in C∪ {∞}, and may be no conjugate. The

exponents γland βjare all non-negative real numbers, and they are the exponents

of zeros zland poles pjof Eq(1.1).

Lemma 1.2. After the coincident and ﬁnite zeros and poles of Eq(1.1) are can-

celled, for any ﬁnite zero of Eq(1.1), its Kvalue is K= +∞; for any ﬁnite pole of

Eq(1.1), its Kvalue is K= 0.

Proof. The Eq(1.1) can be transformed to the following characteristic equation.

KG(s)Qm

l=1(1 −s

zl)γlGlz (s)−(a+ib)Qn

j=1(1 −s

pj)βjGjp (s) = 0. Obviously, if

K= 0, all roots of the pole factor Qn

j=1(1 −s

pj)βjGjp (s) are the roots of charac-

teristic equation. Conversely, all roots of characteristic equation if K= 0 are all

roots of the pole factor. So, at poles of Eq(1.1), K= 0.

The Eq(1.1) can be transformed to its another characteristic equation. G(s)Qm

l=1(1−

s

zl)γlGlz (s)−a+ib

KQn

j=1(1 −s

pj)βjGjp (s) = 0. If K= +∞, all roots of the zero fac-

tor Qm

l=1(1 −s

zl)γlGlz (s) are the roots of the characteristic equation. Conversely,

all roots of the characteristic equation when K= +∞are the roots of the zero

factor. So, at zeros of Eq(1.1), K= +∞.

ZEROS OF MEROMORPHIC FUNCTION 3

According to the expression of Kof the root locus equation (1.1) and according

to Lemma 1.2, it is obvious that the Kvalues are continuous concerning the complex

variable sin C∪ {∞}. So, we can prove Theorem 1.3 simply.

Theorem 1.3. Let s∈C∪ {∞}, the Kvalue of s∈C∪ {∞} takes all of the

non-negative real number value from 0 to the +∞.

For the argument[8] in complex analysis, in automatic control theory, it is called

as the phase angle. Kis called as the gain K.

Let ∆ = {s=σ+it ∈C∪ {∞},sis not zero or pole of W(s) and Eq(1.1)}.

Lemma 1.4. For any point s∈∆, the phase angle of meromorphic function W(s)

can be written as the ϕ(σ, t)function: ϕ(σ, t) = arg(Gy(σ,t)

Gx(σ,t)) + Pm

l=1(γlarg( t−tl

σ−σl)−

γlarg( tl

σl) + arg( Gz y(σ,t)

Gzx(σ,t))) −Pn

j=1(βjarg( t−tj

σ−σj)−βjarg( tj

σj) + arg( Gpy (σ,t)

Gpx(σ,t))).

Let s= (σ, t)∈∆, then the phase angle condition equation of Eq(1.1) on sis:

(1.2) ϕ(σ, t) = 2qπ +arg(b/a)

In which, qis an integer number.

Lemma 1.5. Let Eq(1.1) be the root locus equation of 2qπ +αdegree, for any point

s= (σ, t)∈∆, if the point ssatisﬁes the phase angle condition equation (1.2) of

2qπ +αdegree, then it must be on the root locus of 2qπ +αdegree of Eq(1.1).

Proof. Assuming that a point s1= (σ1, t1)∈∆ satisﬁes the phase angle condition

equation (1.2), ϕ(σ1, t1) = 2qπ +arg(b1/a1). The phase angle of W(s) is ϕ(σ, t),

this phase angle expression is same as the expression of the left side of Eq(1.2).

Because the point s1satisﬁes Eq(1.2), when the point s1is substituted into

the function W(s), W(s) obtains a complex value, according to the phase angle

expression of Eq(1.2) of the point s1and the condition which a point s1satisﬁes

the phase angle condition equation (1.2), this complex value can be written as:

K∗

1(a1+ib1), and α1= arg (b1/a1), K∗

1is the modulus of function W(s) of point

s1.K∗

1is a non-zero positive real value. α1is the phase angle of the function W(s)

of the point s1, and it is the value of the right side of Eq(1.2).

If we bring the point s1into the gain expression |Qn

j=1(1−s

pj)βjGjp (s)

G(s)Qm

l=1(1−s

zl)γlGlz(s)|, we can

obtain an unique gain K1,K1=|Qn

j=1(1−s1

pj)βjGjp(s1)

G(s1)Qm

l=1(1−s1

zl)γlGlz(s1)|, the gain K1is a recipro-

cal of the modulus K∗

1of W(s) of the point s1. Further obtain K∗

1.K∗

1= 1/K1. For

that the gain K1multiplied the expression G(s1)Qm

l=1(1 −s1

zl)γlGlz (s1)/Qn

j=1(1 −

s1

pj)βjGjp (s1), we have, K1G(s1)Qm

l=1(1 −s1

zl)γlGlz (s1)/Qn

j=1(1 −s1

pj)βjGjp (s1) =

a1+ib1. This equation is a concrete situation which a point s1satisﬁes Eq(1.1).

The previous results prove that the point s1is a root of Eq(1.1).

So, the point s1satisﬁes Eq(1.1). Therefore, it is proved that the point s1is

on the root locus of Eq(1.1), and which its gain is K1and its degree is 2qπ +α1.

Hence, if the point s1satisﬁes Eq(1.2), then the point s1is on the root locus of

Eq(1.1), and which its gain is K1and its degree is 2qπ +α1.

Lemma 1.6. Let Eq(1.1) be the root locus equation of 2qπ +αdegree, for any point

s= (σ, t)∈∆, if the point sis on the root locus of 2qπ +αdegree of Eq(1.1), then

it must satisfy the phase angle condition equation of 2qπ +αdegree of Eq(1.2).

4 LANDE MA AND ZHAOKUN MA

Proof. Assume that the point s2= (σ2, t2)∈∆ is an arbitrary point on the root

locus of Eq(1.1). The points which satisfy Eq(1.1) are all on the root locus of

Eq(1.1). So, the point s2surely satisﬁes Eq(1.1). When s2is substituted into

Eq(1.1), K2G(s2)Qm

l=1(1 −s2

zl)γlGlz (s2)/Qn

j=1(1 −s2

pj)βjGjp (s2) = a2+ib2. Ac-

cording to Lemma 1.3, here, the gain K2is a positive real number, its phase angle

of K2is 2q1π, the phase angle of the right side of the equation in this paragraph

on the point s2is 2q2π+ arg (b2/a2) = 2q2π+α2.

The factor of the left side of the equation in last paragraph on the point s2can be

looked as two factors. One is K2, another is G(s2)Qm

l=1(1 −s2

zl)γlGlz (s2)/Qn

j=1(1 −

s2

pj)βjGjp (s2). The phase angle of the factor of the left side of the equation in last

paragraph on the point s2is equal to the summation of two phase angles of two

factors of K2and G(s2)Qm

l=1(1 −s2

zl)γlGlz(s2)/Qn

j=1(1 −s2

pj)βjGjp (s2). The phase

angle of the factor of the left side of the equation in last paragraph on the point s2

is equal to the phase angle of the right side of the equation in the last paragraph

on the point s2. The phase angle 2q2π+α2subtracts the phase angle 2q1πis equal

to 2q2π+α2−2q1π= 2qπ +α2.

The diﬀerence of the phase angle of the factor a2+ib2and the factor K2is:

2qπ +α2. So , the phase angle of expression G(s2)Qm

l=1(1 −s2

zl)γlGlz (s2)/Qn

j=1(1 −

s2

pj)βjGjp (s2) is 2qπ +α2= 2qπ + arg (b2/a2). We can obtain: the phase angle of

W(s) is ϕ(σ, t) = arg(Gy(σ,t)

Gx(σ,t)) + Pm

l=1(γlarg( t−tl

σ−σl)−γlarg( tl

σl) + arg( Gz y (σ,t)

Gzx(σ,t))) −

Pn

j=1(βjarg( t−tj

σ−σj)−βjarg( tj

σj) + arg( Gpy (σ,t)

Gpx(σ,t))). So, according to the previous

proof, we can obtain: ϕ(σ2, t2) = 2qπ +arg(b2/a2), the point s2satisﬁes the phase

angle condition equation (1.2) . Hence, if the point s2is on the root locus of Eq(1.1),

which its gain is K2and its degree is 2qπ +α2, and satisﬁes Eq(1.1), then point s2

satisﬁes Eq(1.2).

Lemma 1.5 gives the suﬃcient condition result of Theorem 1.7. Lemma 1.6 gives

the necessary condition result of Theorem 1.7. So, sum up Lemma 1.5 and Lemma

1.6, we can obtain Theorem 1.7.

Theorem 1.7. Let Eq(1.1) be 2qπ +αdegree and s= (σ, t)∈∆be an arbitrary

point. A necessary and suﬃcient condition for that the point sis on the root locus

of 2qπ +αdegree of Eq(1.1) is that point swhether or not satisﬁes the phase angle

condition equation (1.2).

Deﬁnition 1.8. Let Ξ be a point set in C∪ {∞}, that are consist of all of points

on the path of roots of Eq(1.1) traced out in C∪ {∞} as 2qπ +α= 2qπ + arg( b

a) is

a constant. That path of Eq(1.1) in C∪ {∞} is called as the root locus of Eq(1.1).

Namely, the set Ξ is the root locus of the 2qπ +αdegree.

Deﬁnition 1.9. When the phase angle of meromorphic function W(s) at the left

side of the root locus equation (1.1) is 2qπ+αdegree, we call the root locus equation

(1.1) as the 2qπ +αdegree root locus equation.

In C∪{∞}, the phase angle of meromorphic function W(s) at the left side of the

root locus equation (1.1) is the degree of the root locus of the root locus equation

(1.1). So, we can give a deﬁnition of the degree of the root locus of Eq(1.1).

Deﬁnition 1.10. When the phase angle of meromorphic function W(s) at the left

side of the root locus equation (1.1) is 2qπ +αdegree, we call the root locus of the

root locus equation (1.1) as 2qπ +αdegree root locus.

ZEROS OF MEROMORPHIC FUNCTION 5

Lemma 1.11. For all ﬁnite zeros of Eq(1.1), they are on the root locus of all

degrees of Eq(1.1) from 2qπ degree to 2qπ + 2γlπdegree.

Proof. Assuming that the point zp is an arbitrary ﬁnite zeros of Eq(1.1). When the

point zp is substituted into meromorphic function W(s) at the left side of Eq(1.1),

we have: G(zp)Qm

l=1(1−z p

zl)γlGlz(z p)

Qn

j=1(1−z p

pj)βjGjp (zp)= 0. This equation can also be expressed as:

G(zp)Qm

l=1(1−z p

zl)γlGlz(z p)

Qn

j=1(1−z p

pj)βjGjp (zp)=Kzp eiθzp = 0. In this equation, Kzp is the modulus of

the function W(s) and θzp is the phase angle of the function W(s). So, Kzp = 0,

and if θzp = 2qπ +α,Kzp ei(2qπ+α)= 0 ∗(cos(θzp ) + isin(θzp )) = 0 is true, and θz p

is an arbitrary degree from 2qπ degree to 2qπ + 2γlπdegree.

For the point zp that lets the left side of Eq(1.1) obtain 0, no matter what

phase angle it is, since its modulus is 0, cos(θz p) + isin(θzp ) is the non-zero and

non-inﬁnity, the modulus of cos(θzp ) + isin(θz p) is 1. θzp represents an arbitrary

degree, which shows: no matter what value θzp is, there is Kzpei(2qπ+α)= 0 ∗

(cos(θzp) + isin(θzp )) = 0.

This proves: For the ﬁnite zero zp of Eq(1.1), when it is substituted into the

meromorphic function W(s), the phase angle of the complex value of the meromor-

phic function W(s) can be the arbitrary 2qπ +αdegree. According to Deﬁnition

1.10, the phase angle of the meromorphic function W(s) is namely the phase angle

of the root locus of Eq(1.1). Because 2qπ +αis an arbitrary degree, when it obtains

all of degrees, this shows the ﬁnite zeros of Eq(1.1) are simultaneously on the root

locus of all of degrees of Eq(1.1) from 2qπ degree to 2qπ + 2γlπdegree.

Lemma 1.12. For all ﬁnite poles of Eq(1.1), they are on the root locus of all

degrees of Eq(1.1) from 2qπ degree to 2qπ + 2βjπdegree.

Proof. Assuming that the point pz is an arbitrary ﬁnite pole of Eq(1.1). When the

point pz is substituted into the meromorphic function W(s), we can obtain a value

Gpz =G(pz)Qm

l=1(1−pz

zl)γlGlz(pz )

Qn

j=1(1−pz

pj)βjGjp (pz)=∞.

In complex analysis, 0 can be written as: 0 ∗eiθ = 0 ∗(co s θ+isin θ). The points

of non-zero and non-inﬁnity ﬁnite values can also be written as: k∗eiθ =k∗(cos θ+

isin θ). The inﬁnity can also be written as: (+∞)∗eiθ = (+∞)∗(cos θ+isin θ).

In which, (cos θ+isin θ) is a non-zero and non-inﬁnity, and its modulus is 1 of the

unit complex number value. For (+∞)∗eiθ = (+∞)∗(cos θ+isin θ), no matter

what value θ= 2qπ +αobtains on the unit circle in C∪ {∞}, the values of this

expression all obtain the inﬁnity.

The above equation can also be expressed as: Gpz =G(pz)Qm

l=1(1−pz

zl)γlGlz(pz )

Qn

j=1(1−pz

pj)βjGjp (pz)=

Kpzeiθp z =∞. In this equation, Kpz is the modulus of the meromorphic function

W(s) and θpz is the phase angle of the meromorphic function W(s). So, Kpz = +∞,

and if θpz = 2qπ +α,Kpz ∗eiθpz = (+∞)∗(cos θpz +isin θpz ) = ∞is true, and

θpz = 2qπ +αis an arbitrary degree from 2qπ degree to 2qπ + 2βjπdegree.

For the point pz that lets the left side of Eq(1.1) obtain ∞, no matter what

phase angle it is, since its modulus is ∞,cos(θpz ) + isin(θpz ) is the non-zero and

non-inﬁnity, the modulus of cos(θpz ) + isin(θpz ) is 1. θpz = 2qπ +αrepresents an

arbitrary degree, which shows: no matter what value θpz = 2qπ +αis, there is

Kpzeiθp z = (+∞)∗(cos(θpz ) + isin(θpz)) = ∞.

6 LANDE MA AND ZHAOKUN MA

For the ﬁnite pole pz of Eq(1.1), when it is substituted into the meromorphic

function W(s), the phase angle of the complex value of the meromorphic function

W(s) can be 2qπ +αdegree. According to Deﬁnition 1.10, the phase angle of the

meromorphic function W(s) is namely the phase angle of the root locus of Eq(1.1).

Because 2qπ +αis an arbitrary degree from 2qπ degree to 2qπ + 2βjπdegree, when

it obtains all of degrees, this shows the ﬁnite poles of Eq(1.1) are simultaneously on

the root locus of all of degrees of Eq(1.1) from 2qπ degree to 2q π +2βjπdegree.

Lemma 1.13. All of the root locus of arbitrary 2qπ +αdegree of Eq(1.1) are

originated from poles of Eq(1.1), and are ﬁnally received by zeros of Eq(1.1).

Proof. The arbitrary 2qπ +αdegree root locus of Eq(1.1) are the curves in C∪

{∞}. So, they all need to have their own origination points and receiving points.

According to the relationship between the points in C∪ {∞} and the root locus

of Eq(1.1), the points in C∪ {∞} can be divided into four types, one is the poles

of Eq(1.1), one is the zeros of Eq(1.1), the other is the general ﬁnite points on the

2qπ +αdegree root locus of Eq(1.1), and the last type is the inﬁnity points in

C∪ {∞}.

Except the ﬁnite and inﬁnite zeros of Eq(1.1), and except the ﬁnite and inﬁnite

poles of Eq(1.1), a general ﬁnite point in C∪ {∞} is on a root loci of a certain

degree of Eq(1.1). And a general ﬁnite point in C∪ {∞} only has the phase angles

of only one degree and non-zero ﬁnite gain value, it can not be the originating point

and receiving point of the root locus of Eq(1.1).

When the inﬁnite point in C∪ {∞} let the meromorphic function W(s) obtain

a non-zero ﬁnite value A. The gain values of the meromorphic function W(s) on

the inﬁnite point in C∪ {∞} are the non-zero ﬁnite value |A|. The phase angles

of the meromorphic function W(s) are all equal to 2qπ +arg(A) degree. So, these

inﬁnite points in C∪ {∞} are also the ordinary points of the ﬁnite values with the

gain |A|on the 2qπ +arg(A) degree root locus, they can not originate or receive

the root locus.

When the inﬁnite points in C∪ {∞} are the inﬁnite zeros or poles of Eq(1.1).

On the inﬁnite zero or pole, the gain value is K= +∞or K= 0 respectively.

The ﬁnite poles and zeros of Eq(1.1) are on the root locus of all of degrees of

Eq(1.1), and the inﬁnite number of the diﬀerent degrees root locus are at the same

one point, so, they satisfy the condition that the root locus can be originated or

received. On the ﬁnite and inﬁnite poles of Eq(1.1), there is K= 0. On the ﬁnite

and inﬁnite zeros of Eq(1.1), there is K= +∞. Thus, we can let the ﬁnite and

inﬁnite poles of Eq(1.1) as the origination points of the root locus. The ﬁnite and

inﬁnite zeros of Eq(1.1) are the receiving points of the root locus.

Deﬁnition 1.14. In C∪ {∞}, if an angle φis between the positive direction of

the real axis and the tangent line of the root loci of arbitrary 2qπ +αdegree that

is originated from a ﬁnite pole, then angle φis called as angle of origination of the

root loci of 2qπ +αdegree of Eq(1.1).

Deﬁnition 1.15. In C∪ {∞}, if an angle ϕis between the positive direction of

the real axis and the tangent line of the root loci of arbitrary 2qπ +αdegree that

receives at a ﬁnite zero, then angle ϕis called as angle of receiving of the root loci

of 2qπ +αdegree of Eq(1.1).

ZEROS OF MEROMORPHIC FUNCTION 7

In which, when the rotation is counterclockwise, the obtained angle is the positive

angle, when the rotation is clockwise the obtained angle is the negative angle.

Theorem 1.16. In C∪ {∞}, let Pkbe a βkrepeated ﬁnite poles of Eq(1.1). The

angle of origination from Pkon the root locus of arbitrary 2qπ +αdegree of Eq(1.1)

is:

θPk=1

βk(2qπ−α+arg(G(Pk))+βkarg(−pk)−arg(Gkp (Pk))+

m

P

l=1

(γlarg(Pk−zl)−

γlarg(−zl) + arg(Glz (Pk)))−

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj)+arg(Gjp (Pk))).

In which, βkis a real number.

Proof. Selecting a point s1on the root locus of 2qπ +αdegree, the point s1should

be inﬁnitely approaching to βkrepeated ﬁnite pole Pkthat needs to compute its

angle of origination. Because point s1is inﬁnitely approaching to the ﬁnite pole

Pk, the phase angle βkarg(s1−pj) of vectors between the point s1and poles (pj,

j6=k. Namely, except pole Pk.) all can be substituted by angles βkarg(Pk−pj),

j6=k. The phase angle γkarg(s1−zl) of vectors between the point s1and all of

the ﬁnite zeros zlof Eq(1.1) all can be substituted by angles γkarg(Pk−zl).

The point s1is on the root locus of 2qπ +αdegree, because this point is inﬁnitely

approaching to the ﬁnite pole Pk, the point s1is inﬁnitely approaching to the

tangent line of the root loci of arbitrary 2qπ +αdegree that originates from the

ﬁnite pole Pk, so, the vector between point s1and the ﬁnite pole Pkis inﬁnitely

approaching to the tangent line of the root loci of arbitrary 2qπ +αdegree that

originates from the ﬁnite pole Pk. So, according to the Deﬁnition 1.14, we can

let the phase angle of vectors between point s1and the ﬁnite pole Pkbe inﬁnitely

approaching to angle of origination θPk.

The point s1must satisfy the phase angle condition equation, arg(G(s1)) +

m

P

l=1

(γlarg(s1−zl)−γlarg(−zl)+ arg(Glz (s1)))−

n

P

j=1

(βjarg(s1−pj)−βjarg(−pj)+

arg(Gj p (s1))) = 2qπ +α. So, when the ﬁnite pole Pksubstitutes point s1, we

can obtained. arg(G(Pk)) +

m

P

l=1

(γlarg(Pk−zl)−γlarg(−zl) + arg(Glz (Pk))) −

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj) + arg(Gjp (Pk))) −βkθPk+βkarg(−pk)−

arg(Gkp (Pk)) = 2qπ +α.

After terms are moved, obtain: βkθPk=arg(G(Pk)) +

m

P

l=1

(γlarg(Pk−zl)−

γlarg(−zl)+arg(Glz (Pk)))−

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj)+arg(Gjp(Pk)))+

βkarg(−pk)−arg(Gkp (Pk)) −2qπ −α

The 2qπ and −2qπ are equivalent, so, 2qπ is substituted by −2qπ, after trans-

positions, we can obtain the formula in Theorem 1.16.

Theorem 1.17. In C∪ {∞}, let Zkbe a γkrepeated zeros of Eq(1.1). The angle

of receiving by Zkon the root locus of arbitrary 2qπ +αdegree of Eq(1.1) is:

θZk=1

γk(2qπ+α−arg(G(Zk))+γkarg(−zk)−arg(Gkz (Zk))−

m

P

l=1,l6=k

(γlarg(Zk−zl)−

γlarg(−zl) + arg(Glz (Zk))) +

n

P

j=1

(βjarg(Zk−pj)−βjarg(−pj) + arg(Gjp (Zk))).

8 LANDE MA AND ZHAOKUN MA

In which, γkis a real number.

Proof. Selecting a point s1on the root locus of 2qπ +αdegree. The point s1is

inﬁnitely approaching to the γkrepeated zero Zkthat needs to compute its angle

of receiving. Because the point s1is inﬁnitely approaching to zero Zk. The phase

angle βkarg(s1−pj) of vectors between point s1and all poles pjof Eq(1.1) all

can be substituted by angles βkarg(Zk−pj). The phase angle γkarg(s1−zl) of

vectors between point s1and all zeros (zl,l6=k, except zero Zk) of Eq(1.1) all can

be substituted by angles γkarg(Zk−zl).

The point s1is on the root locus of 2qπ +αdegree, because this point is inﬁnitely

approaching to the ﬁnite zero Zk, the point s1is inﬁnitely approaching to the

tangent line of the root loci of arbitrary 2qπ +αdegree that is received by the

ﬁnite zero Zk, so, the vector between point s1and the ﬁnite zero Zkis inﬁnitely

approaching to the tangent line of the root loci of arbitrary 2qπ +αdegree that

is received by the ﬁnite zero Zk. So, according to the Deﬁnition 1.15, we can let

the phase angle of vectors between point s1and the ﬁnite zero Zkbe inﬁnitely

approaching to angle of receiving θZk.

The point s1must satisfy the phase angle condition equation, arg(G(s1)) +

m

P

l=1

(γlarg(s1−zl)−γlarg(−zl)+ arg(Glz (s1)))−

n

P

j=1

(βjarg(s1−pj)−βjarg(−pj)+

arg(Gj p (s1))) = 2qπ+α. So, when the ﬁnite zero Zksubstitutes point s1, we can ob-

tained. arg(G(Zk)) +

m

P

l=1,l6=k

(γlarg(Zk−zl)−γlarg(−zl) + arg(Glz (Zk)))+ γkθZk−

γkarg(−zk) + arg(Gkz (Zk)) −

n

P

j=1

(βjarg(Zk−pj)−βjarg(−pj) + arg(Gjp (Zk))) =

2qπ +α.

After terms are moved, obtain: γkθZk=−arg(G(Zk)) −

m

P

l=1,l6=k

(γlarg(Zk−zl)−

γlarg(−zl) + arg(Glz (Zk))) +

n

P

j=1

(βjarg(Zk−pj)−βjarg(−pj) + arg(Gjp (Zk))) +

γkarg(−zk)−arg(Gkz (Zk)) + 2qπ +α. After transpositions, we can obtain the

formula in Theorem 1.17.

Theorem 1.18. The diﬀerence of degrees of the root locus that a pole emits on two

opposite directions is α1−α2=βkπ. In which, α1is the degree that a pole emits

the root locus on one direction, α2is the degree that the pole emits the root locus

on the opposite direction, βkis the exponent of the pole.

Proof. According to Theorem 1.16, two angles of departure at the βkrepeated pole

Pkon two opposite directions are:

θPk=1

βk(2qπ−α1+arg(G(Pk))+βkarg(−pk)−arg(Gkp (Pk))+

m

P

l=1

(γlarg(Pk−zl)−

γlarg(−zl) + arg(Glz (Pk)))−

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj)+arg(Gjp (Pk))).

θPk+π=1

βk(2qπ−α2+arg(G(Pk))+βkarg(−pk)−arg(Gkp (Pk))+

m

P

l=1

(γlarg(Pk−zl)−

γlarg(−zl) + arg(Glz (Pk)))−

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj)+arg(Gjp (Pk))).

ZEROS OF MEROMORPHIC FUNCTION 9

Two sides of two equations separately subtract, and obtain: −π=1

βk(−α1+α2).

−πβk=−α1+α2. So, α1−α2=βkπ.

Theorem 1.19. The diﬀerence of degrees of the root locus that a zero receives on

two opposite directions is α2−α1=γkπ, In which, α1is the degree that a zero

receives the root locus on one direction, α2is the degree that the zero receives the

root locus on the opposite direction, γkis the exponent of zero.

Proof. According to Theorem 1.17, two angles of arrival at the γkrepeated zero Zk

on two opposite directions are:

θZk=1

γk(2qπ+α1−arg(G(Zk))+γkarg(−zk)−arg(Gkz (Zk))−

m

P

l=1,l6=k

(γlarg(Zk−zl)−

γlarg(−zl) + arg(Glz (Zk))) +

n

P

j=1

(βjarg(Zk−pj)−βjarg(−pj) + arg(Gjp (Zk))).

θZk+π=1

γk(2qπ+α2−arg(G(Zk))+γkarg(−zk)−arg(Gkz (Zk))−

m

P

l=1,l6=k

(γlarg(Zk−zl)−

γlarg(−zl) + arg(Glz (Zk))) +

n

P

j=1

(βjarg(Zk−pj)−βjarg(−pj) + arg(Gjp (Zk))).

Two sides of two equations separately subtract, and obtain: −π=1

γk(α1−α2).

−πγk=α1−α2. So, α2−α1=γkπ.

Theorem 1.20. For the root locus which are originated from an arbitrary ﬁnite

pole Pkof Eq(1.1) in C∪ {∞}, if the angle θPkof origination is a independent

variable, then, the degree values of the root locus strictly and monotonously increase

clockwise.

Proof. If the angle of origination of the α1degree root locus of Eq(1.1) on the ﬁ-

nite pole Pkis: θPk1=1

βk(2qπ −α1+arg(G(Pk)) + βkarg(−pk)−arg(Gkp(Pk)) +

m

P

l=1

(γlarg(Pk−zl)−γlarg(−zl)+arg(Glz (Pk)))−

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj)+

arg(Gj p (Pk))). If the angle of origination of the α2degree root locus of Eq(1.1) on

the ﬁnite pole is: θPk2=1

βk(2qπ −α2+arg(G(Pk)) + βkarg(−pk)−arg(Gkp (Pk)) +

m

P

l=1

(γlarg(Pk−zl)−γlarg(−zl)+arg(Glz (Pk)))−

n

P

j=1,j6=k

(βjarg(Pk−pj)−βjarg(−pj)+

arg(Gj p (Pk))). Two equations subtract, obtain: θPk1−θPk2=1

βk(α2−α1), so,

α2−α1=βk(θPk1−θPk2).

βkis the exponent of pole Pk, it is a real number. θPk1and θPk2are separately

the origination angle of the root locus of α1degree and α2degree. The angle

of origination is deﬁned as a angle which is between the positive direction of the

real axis and the tangent line of the root loci that is originated from a ﬁnite pole.

Beginning at the positive real axis, rotating counterclockwise to the tangent line of

the root loci, a angle of origination can be obtained. So, if θPk1> θPk2,θPk1−θPk2>

0. The angle of origination of the root loci of α1degree is θPk1. The angle of

origination of the root loci of α2degree is θPk2.α2−α1>0, α2> α1is true. So,

it can prove: for two root locus which are originated on the two diﬀerent angles of

origination from a same pole, the root loci of the larger degree is at the clockwise

direction of the root loci of the less degree.

If θPk1< θPk2,θPk1−θPk2<0. α2−α1<0, α2< α1is true. So, it can prove:

for two root locus which are originated on the two diﬀerent angles of origination

10 LANDE MA AND ZHAOKUN MA

from a same pole, the root loci of the larger degree is at the clockwise direction of

the root loci of the less degree.

So, the degree values of the root locus that are originated from this pole Pk

strictly and monotonously increases clockwise.

Theorem 1.21. For the root locus which are received by an arbitrary ﬁnite zero

Zkof Eq(1.1) in C∪ {∞}, if the angle θZkof receiving is a independent variable,

then its degree values strictly and monotonously decrease clockwise.

Let the diﬀerence of the degree numbers of a pair of the root locus which begin

at pole Pkon opposite direction be a positive, according to Theorem 1.18, we can

obtain Theorem 1.22. Let the diﬀerence of the degree numbers of a pair of the

root locus which begin at pole Zkon opposite direction be a positive, according to

Theorem 1.19, we can obtain Theorem 1.23.

Theorem 1.22. A necessary and suﬃcient condition for the βkrepeated pole Pk

to be simple is that the diﬀerence of the degree numbers of any one pair of the root

locus which begin at pole Pkon opposite direction is a positive integer times of π.

Theorem 1.23. A necessary and suﬃcient condition for the γkrepeated zero Zk

to be simple is that the diﬀerence of the degree number of any one pair of the root

locus which end at zero Zkon opposite direction is a positive integer times of π.

ξ(s) = 1

2s(s−1)Γ( s

2)π−s

2ζ(s) is a conjugate function, namely, if ξ(s) = u(σ+

it) + iv(σ+it), then, ξ(s) = u(σ+it)−iv(σ+it). ξ(s) = ξ(1 −s). So, ξ(1

2+it) =

ξ(1

2−it), namely, if ξ(1

2+it) = u(1

2+it) + iv(1

2+it), then, ξ(1

2+it) = ξ(1

2−

it) = u(1

2+it) + iv(1

2+it). By the property of the conjugate function, we have:

ξ(1

2−it) = u(1

2+it)−iv(1

2+it). So, v(1

2+it) = 0, on the critical line, xi-function

obtains the real number values.

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Current address: School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

Email address:dzy200408@126.com

Current address: YanZhou College, ShanDong Radio and TV University, YanZhou, ShanDong

272100 China

Email address:dzy200408@sina.cn