Complex Riccati differential equations revisited

Steinmetz, Norbert

doi:10.5186/aasfm.2014.3929

Cited by 14 publications

(15 citation statements)

References 12 publications

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“…We quote from that the solutions to have asymptotic expansions

t false(z false) \sim z^{\frac{n}{2}} (ε + \sum_{k = 1}^{\infty} c_{k} (ε) z^{- \frac{k}{2}}) false(ε = ε_{ν} (t) \in {- 1, 1} false)

on the Stokes sectors

{normalΣ}_{ν} : false| prefixarg z - \frac{2 ν π}{n + 2} false| < \frac{π}{n + 2}

false| z false| > r_{0}

(

0 ⩽ ν ⩽ n + 1

); for

n

, even the coefficients with odd index vanish. The solution

sans-serift

is uniquely determined if the asymptotic expansion holds on some open sector that contains

{\bar{normalΣ}}_{ν}

; it then holds on

{normalΣ}_{ν - 1} \cup σ_{ν - 1} \cup {normalΣ}_{ν} \cup σ_{ν} \cup {normalΣ}_{ν + 1}

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…They arise from the zeros of

{sans-serifr}_{2} false(z, sans-serift (z) false)

, and also from the poles of

sans-serift

{prefixdeg}_{t} {sans-serifr}_{1} > {prefixdeg}_{t} {sans-serifr}_{2}

, hence

{prefixdeg}_{t} {sans-serifr}_{1} = 1 + {prefixdeg}_{t} {sans-serifr}_{2}

. Regarding the poles of

sans-serift

we recall some facts from . Up to finitely many, the poles of any generic solution are arranged in

n + 2

sequences

(p_{k})

satisfying the approximate iteration scheme

p_{k + 1} = p_{k} \pm false(π i + o (1) false) p_{k}^{- n / 2},

hence

p_{k} = {(() (false(\frac{n}{2} + 1 false) π i + o false(1 false)) k)}^{2 / (n + 2)}

, with counting function

n false(r, (p_{k}) false) = \frac{r^{false(n / 2 false) + 1}}{(n + 2) π} + o false(r^{false(n / 2 false) + 1} false) .

To each Stokes ray

σ_{ν} : arg z = θ_{ν} = \frac{2 ν + 1}{n + 2} π

there is exactly one such string that is asymptotic to

σ_{ν}

…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Example The solutions to belong to the class

{\tilde{frakturY}}_{n / 2, n / 2}

, the components of the solutions to the Hamiltonian system

p^{'} = - q^{2} - z p - a

q^{'} = p^{2} + z q + b

belong to

{\tilde{frakturY}}_{1, 1}

, and the first, second, and fourth Painlevé transcendents belong to the classes

{\tilde{frakturY}}_{\frac{1}{2}, \frac{1}{4}}

{\tilde{frakturY}}_{\frac{1}{2}, \frac{1}{2}}

, and

{\tilde{frakturY}}_{1, 1}

, respectively (see ).

◇

…”

Section: Re‐scalingmentioning

confidence: 99%

“…They arise from the zeros of r 2 (z, t(z)), and also from the poles of t if deg t r 1 > deg t r 2 , hence deg t r 1 = 1 + deg t r 2 . Regarding the poles of t we recall some facts from [15]. Up to finitely many, the poles of any generic solution are arranged in n + 2 sequences (p k ) satisfying the approximate iteration scheme…”

Section: The Distribution Of Polesmentioning

confidence: 99%

“…Remark 7. The re-scaling method was introduced in [14][15][16] in the context of various analytic differential equations. It was inspired by the well-known Zalcman Re-scaling Lemma [22,23] and Yosida's work [19].…”

Section: A Normality Criterionmentioning

confidence: 99%

See 4 more Smart Citations

First-order algebraic differential equations of genus zero

Steinmetz

2017

Bull. London Math. Soc.

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“…We quote from that the solutions to have asymptotic expansions

t false(z false) \sim z^{\frac{n}{2}} (ε + \sum_{k = 1}^{\infty} c_{k} (ε) z^{- \frac{k}{2}}) false(ε = ε_{ν} (t) \in {- 1, 1} false)

on the Stokes sectors

{normalΣ}_{ν} : false| prefixarg z - \frac{2 ν π}{n + 2} false| < \frac{π}{n + 2}

false| z false| > r_{0}

(

0 ⩽ ν ⩽ n + 1

); for

n

, even the coefficients with odd index vanish. The solution

sans-serift

is uniquely determined if the asymptotic expansion holds on some open sector that contains

{\bar{normalΣ}}_{ν}

; it then holds on

{normalΣ}_{ν - 1} \cup σ_{ν - 1} \cup {normalΣ}_{ν} \cup σ_{ν} \cup {normalΣ}_{ν + 1}

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…They arise from the zeros of

{sans-serifr}_{2} false(z, sans-serift (z) false)

, and also from the poles of

sans-serift

{prefixdeg}_{t} {sans-serifr}_{1} > {prefixdeg}_{t} {sans-serifr}_{2}

, hence

{prefixdeg}_{t} {sans-serifr}_{1} = 1 + {prefixdeg}_{t} {sans-serifr}_{2}

. Regarding the poles of

sans-serift

we recall some facts from . Up to finitely many, the poles of any generic solution are arranged in

n + 2

sequences

(p_{k})

satisfying the approximate iteration scheme

p_{k + 1} = p_{k} \pm false(π i + o (1) false) p_{k}^{- n / 2},

hence

p_{k} = {(() (false(\frac{n}{2} + 1 false) π i + o false(1 false)) k)}^{2 / (n + 2)}

, with counting function

n false(r, (p_{k}) false) = \frac{r^{false(n / 2 false) + 1}}{(n + 2) π} + o false(r^{false(n / 2 false) + 1} false) .

To each Stokes ray

σ_{ν} : arg z = θ_{ν} = \frac{2 ν + 1}{n + 2} π

there is exactly one such string that is asymptotic to

σ_{ν}

…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Example The solutions to belong to the class

{\tilde{frakturY}}_{n / 2, n / 2}

, the components of the solutions to the Hamiltonian system

p^{'} = - q^{2} - z p - a

q^{'} = p^{2} + z q + b

belong to

{\tilde{frakturY}}_{1, 1}

, and the first, second, and fourth Painlevé transcendents belong to the classes

{\tilde{frakturY}}_{\frac{1}{2}, \frac{1}{4}}

{\tilde{frakturY}}_{\frac{1}{2}, \frac{1}{2}}

, and

{\tilde{frakturY}}_{1, 1}

, respectively (see ).

◇

…”

Section: Re‐scalingmentioning

confidence: 99%

Section: The Distribution Of Polesmentioning

confidence: 99%

Section: A Normality Criterionmentioning

confidence: 99%

See 3 more Smart Citations

First-order algebraic differential equations of genus zero

Steinmetz

2017

Bull. London Math. Soc.

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A note on properties of solutions of Riccati and hyper‐Riccati equations

Ciechanowicz

2019

Z Angew Math Mech

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The paper focuses on transcendental meromorphic solutions of Riccati and hyper-Riccati equations and their value distribution properties. Special attention is paid to deficient values in various senses, in particular to deficient values in the sense of Petrenko in the case of solutions of finite order. The question of small deficient functions is also considered. The estimates of deficiencies, multiplicity indices and, in the case of functions of finite order, also deviations from small functions are presented. K E Y W O R D S meromorphic function, Riccati equation, value distribution M S C ( 2 0 1 0 ) 30D35, 34M05

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Selected Topics in Complex Analysis

Steinmetz

2017

Universitext

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Complex Riccati differential equations revisited

Abstract: Abstract. We utilise a new approach via the so-called re-scaling method to derive a thorough theory for polynomial Riccati differential equations in the complex domain.

Cited by 14 publications

References 12 publications

First-order algebraic differential equations of genus zero

First-order algebraic differential equations of genus zero

A note on properties of solutions of Riccati and hyper‐Riccati equations

Selected Topics in Complex Analysis

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