TOWARDS AN EFFECTIVISATION OF THE RIEMANN THEOREM

S.Natanzon

Independent University of Moscow, Institute Theoretical and Experimental Physics

Abstract

The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map for a broad class of domains with analytic boundaries.

1. Introduction.

The Riemann Theorem states, that if connected and simply connected subdomain Q on the Riemann sphere ℂ̄ is not nonequivalent to complex plane, then there exists a Riemann map. The Riemann map is a conformal bijection w_{Q} between Q and the exterior of the unit disk U = {u ∈ ℂ̄||u| > 1}. In this paper we find an explicit form of the Riemann map for a broad class of domains with analytic boundaries.

Without loss of generally it is possible to consider that ∞ ∈ Q and ∞ ∈ Q_{∘}, where Q_{∘} = ℂ̄ − (Q ∪ ∂Q) and ∂Q is the boundary of Q. Let ℍ be the set of all such domains. The harmonic moments

t_{0} = \frac{1}{π} ∫_{Q_{∘}} dxdy, t_{k} = −\frac{1}{πk} ∫_{Q} z^{−k}dxdy (k = 1, 2...)  (1.1)

form a coordinate system t = (t_{0}, t_{1}, t_{2}, ...) on a considerable part of ℍ [ 1, 13 ]. Let Q^{t} ∈ ℍ be the domain corresponding to t. The conditions ∂_{z}w_{Q^{t}}(∞) ∈ ℝ, ∂_{z}w_{Q^{t}}(∞) > 0 uniquely define the Riemann maps

w(z, t) = w_{Q^{t}}(z) = p(t)z + ∑_{j=0}^{∞} p_{j}(t)z^{−j},  p(t) ∈ ℝ, p(t) > 0.  (1.2)

Our goal is a presentation p_{j}(t) as Taylor series from t and t̄.

Put

∂_{i} = \frac{∂}{∂t_{i}} , ¯∂_{i} = \frac{∂}{∂t̄_{i}} , D(z) = ∑_{k⩾1} \frac{z^{−k}}{k} ∂_{k}, D̄(z̄) = ∑_{k⩾1} \frac{z̄^{−k}}{k} ¯∂_{k}  (1.3)

Typeset by AMS-TEX

In §1 we, following by [ 2,14,15 ], reproduce an important result [ 3 ]: the Riemann maps have a potential F : ℍ → ℂ (tau-function of the curves as regards to [10]) suth that

w(z, t) = zexp((−\frac{1}{2}∂_{0}^{2} − ∂_{0}D(z))F (t))  (1.4)

and moreover F satisfies differential equations

(z − ξ)e^{D(z)D(ξ)F} = ze^{−∂_{0}D(z)F} − ξe^{−∂_{0}D(ξ)F} .  (1.5(a))
(z̄ − ξ̄)e^{D̄(z̄)D̄(ξ̄)F} = z̄e^{−∂_{0}D̄(z̄)F} − ξ̄e^{−∂_{0}D̄(ξ̄)F} .  (1.5(b))
1 − e^{−D(z)D̄(ξ̄)F} = \frac{1}{zξ̄}e^{∂_{0}(∂_{0}+D(z)+D̄(ξ̄))F} ,  (1.5(c))

This system of the nonlinear differential equations is well known in mathematical physics and in theory of integrable systems as dispersionless limit of the 2D Toda hierarchy [ 4 ]. The solution F (t) satisfies some additional equation, and it appears in string theory as ”string solution” [ 5 ]. The string solution of dispersionless limit of 2D Toda hierarchy appears also in matrix models and in some other problems of mathematical physics [ 6,13 ]. Thus, a description of it has an independent interest.

In §2 we, follow by [ 7 ], find recurrent formulas for coefficients of the Taylor series for the potential.

F = ∑ N(i|i_{1}..., i_{k}|ī_{1}..., ī_{k})t_{0}^{i}t_{i_{1}} ⋯ t_{i_{k}} t̄ī¯1 ⋯ t̄īk̄.  (1.6)

These formulas together with (1.4) allow to find the Riemann maps via harmonic moments of domains.

In §3 we follow by [ 8 ] find sufficient condition for convergence of the Taylor series (1.6). I acknowledge useful discussion with S.P.Novikov, A.Marshakov, A.Zabrodin. The work

was partially supported by grant RFBR 02-01-22004a and grant INTAS-00-0259.

2. Potential for Riemann maps.

The bijection w : Q → U generates the Dirichlet Green function on Q × Q

G_{Q}(z, ξ) = log |\frac{w(z) − w(ξ)}{w(z)w̄(ξ) − 1}| .  (2.1)

It solves a Dirichlet problem in Q: the formula

u(z) = −\frac{1}{2π} ∮_{∂Q} u_{0}(ξ)∂_{n}G_{Q}(z, ξ)|dξ|  (2.2)

restores a harmonic function by its boundaries values u_{0} = u|_{∂D}. Here ∂_{n} is the normal derivative inward Q on the boundary ∂Q with respect to the second variable, and |dξ| is an infinitesimal element of ligth of ∂Q.

The Dirichlet Green function is uniquely determined by the following properties [ 9 ]. G_{Q}(z, ξ) is symmetric and harmonic everywhere in Q in both arguments except z = ξ,

where G_{Q}(z, ξ) = log |z − ξ| + O(1) as z → ξ. (2.3(a))

G_{Q}(z, ξ) = 0 if both of variables z, ξ belong to the boundary. (2.3(b)) Denote by ℍ_{z} the set of all domains Q ∈ ℍ containing z. The infinitesimal shift of

the boundary ∂Q = \{ξ\} on −\frac{ε}{2}∂_{n}G_{Q}(z, ξ) (ε → 0) generates a vector field δ_{z} on ℍ_{z}. In particulary,

δ_{z}(t_{k}) = −\frac{1}{επk} ∮_{∂Q} ξ^{−k}(−\frac{ε}{2}∂_{n}G_{Q}(z, ξ))|dξ| = \frac{z^{k}}{k} (k > 0)  δ_{z}(t_{0}) = 1. (2.4)

Thus, for any functional X : ℍ_{z} → ℂ

δ_{z}X = \frac{∂X}{∂t_{0}} δ_{z}t_{0} + ∑ \frac{∂X}{∂t_{k}} δ_{z}t_{k} + ∑ \frac{∂X}{∂t̄_{k}} δ_{z}t̄_{k} = ∇(z)X,  (2.5)

where ∇(z) = ∂_{0} + D(z) + D̄(z̄).

Put v_{0} = \frac{2}{π} ∫_{Q_{∘}} log |z|dxdy and v_{k} = \frac{1}{π} ∫_{Q_{∘}} z^{k}dxdy for k > 0.

Theorem 2.1. [3,10,15]. The function F (t) = −\frac{1}{π} ∫∫_{Q_{∘}^{t}Q_{∘}^{t}} log |η^{−1} − ν^{−1}|dx_{η}dy_{η}dx_{ν}dy_{ν}

(where Q_{∘}^{t} = ℂ̄ − Q^{t}) satisfies the equations (1.4) and (1.5). Moreover ∂_{k}F = v_{k}. Proof: It is follow from (2.2) and from the definition of δ_{z} that

G̃_{Q}(z, ξ) = log |\frac{1}{z} − \frac{1}{ξ}| + \frac{1}{2}δ_{z}δ_{ξ}F  (2.6)

satisfies (2.3). Thus, G̃_{Q}(z, ξ) = G_{Q}(z, ξ). Using (2.1) and (2.5), we find that

log |\frac{w(z) − w(ξ)}{w(z)w̄(ξ) − 1}| = log |\frac{1}{z} − \frac{1}{ξ}| + \frac{1}{2}∇(z)∇(ξ)F  (2.7)

This equation implies equations (1.4) and (1.5). Really, it implies

h = log |\frac{w(z) − w(ξ)}{w(z)w̄(ξ) − 1}|^{2} − log |\frac{1}{z} − \frac{1}{ξ}|^{2} − ∇(z)∇(ξ)F = 0.  (2.8)

Besides h = h_{1} + h_{2} where

h_{1} = log (\frac{w(z) − w(ξ)}{w(z)w̄(ξ) − 1}) − log (\frac{1}{z} − \frac{1}{ξ}) − (\frac{1}{2}∂_{0} + D(z)) ∇(ξ)F  (2.9)

is a holomorphic function of z while

h_{2} = log (\frac{w̄(z) − w̄(ξ)}{w̄(z)w(ξ) − 1}) − log (\frac{1}{z̄} − \frac{1}{ξ̄}) − (\frac{1}{2}∂_{0} + D̄(z̄)) ∇(ξ)F  (2.10)

is an antiholomorphic function of z. Thus, h_{1} is independent from z. Passing on to the limit z → ∞, we obtain

h_{1} = log (\frac{1}{w̄(ξ)}) − log (−\frac{1}{ξ}) − \frac{1}{2}∂_{0}∇(ξ)F.  (2.11)

Thus,

log (\frac{w(z) − w(ξ)}{z − ξ} \frac{zw̄(ξ)}{w(z)w̄(ξ) − 1}) = D(z)∇(ξ)F.  (2.12)

Passing on to the limit ξ → ∞ in (2.12), we obtain log (\frac{pz}{w(z)}) = D(z)∂_{0}F. Substituting

(1.2) in (2.7) and passing on to the limit z → ∞, ξ → ∞, we obtain log(p) = −\frac{1}{2}∂_{0}^{2}F. A comparison of these formulas gives

log (\frac{z}{w(z)}) = D(z)∂_{0}F + \frac{1}{2}∂_{0}^{2}F,  (2.13)

that is equivalent (1.4).

Using (2.13) we transform the holomorphic parts of (2.12) to

log (\frac{w(z) − w(ξ)}{z − ξ}) = −\frac{1}{2}∂_{0}^{2}F + D(z)D(ξ)F.  (2.14)

A substituting (1.4) to (2.14) gives (1.5(a)). Changing (z, ξ) to (z̄, ξ̄) in the proof we obtain (1.5(b)).

The antiholomorphic parts of (2.12) is

− log (1 − \frac{1}{w(z)w̄(ξ)}) = D(z)D̄(ξ̄)F.  (2.15)

A substituting (1.4) to (2.15) gives (1.5(c)).

Moreover if z ∈ ∂Q then

∇(z)F = δ_{z}F = −\frac{2}{π} ∫_{Q_{∘}^{t}(t)} log |ν^{−1} − z^{−1}|dνdν̄ = v_{0} + 2Re ∑_{k⩾1} \frac{v_{k}}{k} z^{−k}.

Thus, ∂_{k}F = v_{k}. □

3. Taylor series for the potential.

In this section we follow by [ 7 ] find recurrent formulas for coefficients N of the Taylor series F = ∑ N(i|i_{1}..., i_{k}|ī_{1}..., ī_{k})t_{0}^{i}t_{i_{1}} ⋯ t_{i_{k}} t̄ī¯1 ⋯ t̄īk̄ for the potential F.

The formulas for N are found by the following scheme. At first, using some combinatorial calculations, we transform the equation (1.5(a)) to an infinite system of equations

∂_{i_{1}}∂_{i_{2}} ⋯ ∂_{i_{k}}F =

= ∑_{m=1}^{∞} (  ∑_{\begin{matrix}s_{1}+···+s_{m}=i_{1}+···+i_{k} \\ ℓ_{1}+···+ℓ_{m}=m+k−2 \\ s_{j},ℓ_{j}⩾1 \end{matrix}} \frac{i_{1} ⋯ i_{k}}{s_{1} ⋯ s_{m}} T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ∂_{0}^{ℓ_{1}}∂_{s_{1}}F ⋯ ∂_{0}^{ℓ_{m}}∂_{s_{m}}F ) . (3.1)

In passing we find some recurrent formulas for calculation of T. Later, using the definition of F as a function on the space of analytical curves, we find,

that

∂_{0}F |_{t_{0}} = −t_{0} + t_{0} ln t_{0} + const and ∂_{k}F |_{t_{0}} = 0, if k > 0, where here and later |_{t_{0}} means the restriction of a function on a straight line t_{1} = t̄_{1} = t_{2} = t̄_{2} = ⋯ = 0.

For this formula and from the equation (1.5(c)) follow, that

∂_{i} ¯∂_{j}F |_{t_{0}} = \{ \begin{matrix}0, if i ≠ j, \\ it_{0}^{i}, if i = j . \end{matrix}

Later, using (3.1) and the symmetry of the equations (1.5) we find, that

∂_{i} ¯∂_{i_{1}} ⋯ ¯∂_{i_{k}}F |_{t_{0}} = ¯∂_{i}∂_{i_{1}} ⋯ ∂_{i_{k}}F |_{t=0} = \{ \begin{matrix}0, if i_{1} + ⋯ + i_{k} ≠ i, \\ i_{1} ⋯ i_{k} \frac{i!}{(i−k+1)!}t_{0}^{i−k+1}, if i_{1} + ⋯ + i_{k} = i. \end{matrix}

This condition and the equation (3.1) give some recurrent formulas for coefficients N. As the final result we get

Theorem 3.1.Let the formal series (1.6) satisfies in its convergence domain the equations (1.4) and (1.5). Then up to linear functions it has the form

F = \frac{1}{2} t_{0}^{2} log t_{0} − \frac{3}{4} t_{0}^{2} + ∑_{\begin{matrix}k,k̄,n_{r},n̄_{r}⩾1 \\ 0<i_{1}<···<i_{k} \\ 0<ī_{1}<···<īk̄ \\ i−(n_{1}+···+n_{k}+n̄_{1}+···+n̄k̄)+2⩾0 \end{matrix}} \frac{i_{1}^{n_{1}} … i_{k}^{n_{k}}}{n_{1}! … n_{k}!} \frac{īn̄_{1} ; 1 … īn̄k̄ ; k̄}{n̄_{1}! … n̄k̄!}·

·N_{i}^{2} ( \begin{matrix}i_{1}, & …, & i_{k} \\ n_{1}, & …, & n_{k} \end{matrix} | \begin{matrix}ī_{1}, & …, & īk̄ \\ n̄_{1}, & …, & n̄k̄ \end{matrix} ) t_{0}^{i−(n_{1}+···+n_{k}+n̄_{1}+···+n̄k̄)+2}t_{i_{1}}^{n_{1}} … t_{i_{k}}^{n_{k}} t̄n̄_{1} ; ī_{1} … t̄n̄k̄ ; īk̄

In this formula the coefficients N^{2} are found by follow recurrent rules:

1) P_{i,j}(s_{1}, …, s_{m}) = # {(i_{1}, …, i_{m}) | i = i_{1} + ⋯ + i_{m}, 1 ⩽ i_{r} ⩽ s_{r} − 1}, where #V is the cardinality of the set V;

2) T_{i,j}^{1}(s_{1}, …, s_{m}) =

∑_{\begin{matrix}k⩾1 \\ n_{1}+···+n_{k}=m \\ n_{r}⩾1 \end{matrix}} \frac{1}{kn_{1}!...n_{k}!} P_{i,j} (\underbrace{s_{1} + ⋯ + s_{n_{1}}}_{n_{1}}, …, \underbrace{s_{n_{1}+···+n_{k−1}+1} + ⋯ + s_{n_{1}+···+n_{k}}}_{n_{k}});

T_{i_{1},i_{2}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ 1, & …, & 1 \end{matrix} ) = T_{i_{1},i_{2}}^{1}(s_{1}, …, s_{m});

T_{i_{1},...,i_{k}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) =

6 S.NATANZON

= ∑_{\begin{matrix}1⩽i⩽j⩽m \\ s,l⩾1 \end{matrix}} l T_{s,i_{k}}^{1}(s_{i}, …, s_{j}) T_{i_{1},...,i_{k−1}}^{2} ( \begin{matrix}s_{1}, & …, & s_{i−1}, & s, & s_{j+1}, & …, & s_{m} \\ l_{1}, & …, & l_{i−1}, & l, & l_{j+1}, & …, & l_{m} \end{matrix} ) ,

where s = s_{i} + ⋯ + s_{j} − i_{k}, l = (l_{i} − 1) + ⋯ + (l_{j} − 1) ;

3) S_{ī_{1},...,īk̄} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) =

= ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}=\{ī_{1},...,īk̄\} \\ ī_{r}^{1}+···+ī_{r}^{nr} =s_{r} \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1}−1)!}{(s_{1}−n_{1}−l_{1}+1)!(l_{1}−1)!} × ⋯ × \frac{(s_{m}−1)!}{(s_{m}−n_{m}−l_{m}+1)!(l_{m}−1)!} ;

4) N_{i}^{1}(i_{1}, …, i_{k}|ī_{1}, …, īk̄) = 0, if i ≠ i_{1} + ⋯ + i_{k} or i ≠ ī_{1} + ⋯ + īk̄ ;

in the other cases N_{i}^{1}(i|ī_{1}, …, īk̄) = \frac{(i−1)!}{(i−k̄+1)!} ; N_{i}^{1}(i_{1}, …, i_{k}|ī) = \frac{(i−1)!}{(i−k+1)!} ; N_{i}^{1}(i_{1}, …, i_{k}|ī_{1}, …, īk̄) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+···+i_{k} \\ l_{1}+···+l_{m}=m+k−2 \\ s_{r},l_{r}⩾1 \end{matrix}}(−1)^{m+1} S_{ī_{1},...,īk̄} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) T_{i_{1},...,i_{k}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) , if

k, k̄ > 1 ;

N_{i}^{2} ( \begin{matrix}i_{1}, & …, & i_{k} \\ n_{1}, & …, & n_{k} \end{matrix} | \begin{matrix}ī_{1}, & …, & īk̄ \\ n̄_{1}, & …, & n̄k̄ \end{matrix} ) =

= N_{i}^{1} (\underbrace{i_{1}, …, i_{1}}_{n_{1}}, …, \underbrace{i_{k}, …, i_{k}}_{n_{k}} |\underbrace{ī_{1}, …, ī_{1}}_{n̄_{1}}, …,\underbrace{īk̄, …, īk̄}_{n̄k̄} ) . □

A proof follow by next lemmas

Lemma 3.1. The following relations hold

z − ∑_{j=1}^{∞} \frac{1}{j}z^{−j}∂_{1}∂_{j}F = ze^{−∂_{0}D(z)}F

∂_{1}∂_{j}F = ∑_{m=1}^{∞} \frac{(−1)^{m+1}}{m!} ∑ ; \begin{matrix}k_{1} + ⋯ + k_{m} = j + 1 \\ k_{i} > 0 \end{matrix} \frac{j}{k_{1} ⋯ k_{m}} ∂_{0}∂_{k_{1}}F ⋯ ∂_{0}∂_{k_{m}}F.

TOWARDS AN EFFECTIVISATION OF THE RIEMANN THEOREM 7

Proof: According (1.5(a)), (z−ξ)e^{D(z)D(ξ)F} = (z−ξ)(1+(D(z)D(ξ)F)+ \frac{1}{2}(D(z)D(ξ)F)^{2}+

⋯) = (z − ξ)(1 + z^{−1}ξ^{−1}∂_{1}^{2}F + z^{−1} ∑_{j=2}^{∞} \frac{1}{j}ξ^{−j}∂_{1}∂_{j}F + ξ^{−1} ∑_{j=2}^{∞} \frac{1}{j}z^{−j}∂_{1}∂_{j}F + z^{−2}ξ^{−2}f) =

(z − ξ) + ξ^{−1}∂_{1}^{2}F − z^{−1}∂_{1}^{2}F + ∑_{j=2}^{∞} \frac{1}{j}ξ^{−j}∂_{1}∂_{j}F − ∑_{j=2}^{∞} \frac{1}{j}z^{−j}∂_{1}∂_{j}F + z^{−1}ξ^{−1}f.

On the other hand, according (1.5(a)), the function (z − ξ)e^{D(z)D(ξ)F} is a sum of two functions f_{1}(z) + f_{2}(ξ). Thus, f = 0 and ze^{−∂_{0}D(z)F} = z − ∑_{j=1}^{∞} \frac{1}{j}z^{−j}∂_{1}∂_{j}F. Therefore,

∑_{j=1}^{∞} \frac{1}{j}z^{−(j+1)}∂_{1}∂_{j}F = 1 − e^{−∂_{0}D(z)F} = 1 − (1 + ∑_{m=1}^{∞} \frac{(−∂_{0}D(z)F)^{m}}{m!} ) =

− ∑_{m=1}^{∞} \frac{(−1)^{m}}{m!} (∑_{k=1}^{∞} \frac{z^{−k}}{k} ∂_{0}∂_{k}F )^{m} =

= − ∑_{m=1}^{∞} \frac{(−1)^{m}}{m!} (∑_{n=1}^{∞} z^{−n} ∑_{k_{1}+···+k_{m}=n} \frac{1}{k_{1} … k_{m}} ∂_{0}∂_{k_{1}}F ⋯ ∂_{0}∂_{k_{m}}F ) =

= − ∑_{n=1}^{∞} z^{−n}( ∑_{m=1}^{∞} \frac{(−1)^{m}}{m!} ∑_{k_{1}+···+k_{m}=n} \frac{1}{k_{1} … k_{m}} ∂_{0}∂_{k_{1}}F ⋯ ∂_{0}∂_{k_{m}}F ).

Thus,

\frac{1}{j}∂_{1}∂_{j}F = − ∑_{m=1}^{∞} \frac{(−1)^{m}}{m!} ∑_{k_{1}+···+k_{m}=j+1} \frac{1}{k_{1} … k_{m}} ∂_{0}∂_{k_{1}}F ⋯ ∂_{0}∂_{k_{m}}F.□

Lemma 3.2. The following relation holds

∂_{i}∂_{j}F = ∑_{m=1}^{∞} ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = j + 1 \\ s_{i} > 1 \end{matrix} \frac{(−1)^{m+1}}{m} \frac{ij}{(s_{1} − 1) ⋯ (s_{m} − 1)}P_{ij}(s_{1} − 1, ..., s_{m} − 1)·

·∂_{1}∂_{s_{1}−1}F ⋯ ∂_{1}∂_{s_{m}−1}F,

where P_{ij}(s_{1} − 1, ..., s_{m} − 1) is the number of representations {i = i_{1} + ⋯ + i_{m}|1 ⩽ i_{k} ⩽ s_{k} − 1, k = 1, ⋯, m} for number i.

Proof: According to lemma 3.1 and equation (1.5(a)), (z − ξ)e^{D(z)D(ξ)F} =

= z − ∑_{j=1}^{∞} \frac{1}{j}z^{−j}∂_{1}∂_{j}F − (ξ − ∑_{j=1}^{∞} \frac{1}{j}ξ^{−j}∂_{1}∂_{j}F) = (z − ξ) − ∑_{j=1}^{∞} \frac{1}{j}(z^{−j} − ξ^{−j})∂_{1}∂_{j}F.

Thus,

e^{D(z)D(ξ)} = 1 + z^{−1}ξ^{−1} ∑_{j=1}^{∞} \frac{1}{j} \frac{(z^{−j} − ξ^{−j})}{(z^{−1} − ξ^{−1})}∂_{1}∂_{j}F =

= 1 + z^{−1}ξ^{−1} ∑_{j=1}^{∞} \frac{1}{j}( ∑ ; \begin{matrix}s + t = j − 1 \\ s, t ⩾ 0 \end{matrix} z^{−s}ξ^{−t})∂_{1}∂_{j}F =

= 1 + ∑_{j=1}^{∞} \frac{1}{j}( ∑ ; \begin{matrix}s + t = j + 1 \\ s, t ⩾ 1 \end{matrix} z^{−s}ξ^{−t})∂_{1}∂_{j}F.

Therefore,

D(z)D(ξ)F = ∑_{m=1}^{∞} \frac{(−1)^{m+1}}{m} (∑_{n=1}^{∞}( ∑ ; \begin{matrix}s + t = n + 1 \\ s, t ⩾ 1 \end{matrix} z^{−s}ξ^{−t})\frac{1}{n}∂_{1}∂_{n}F )^{m} =

= ∑_{j=1}^{∞} \frac{(−1)^{m+1}}{m} ∑_{i,j⩾1} z^{−i}ξ^{−j}·

( ∑ ; \begin{matrix}i_{1} + ⋯ + i_{m} = i \\ j_{1} + ⋯ j_{m} = j \\ i_{k}, j_{k} ⩾ 1 \end{matrix} \frac{1}{i_{1} + j_{1} − 1}∂_{1}∂_{i_{1}+j_{1}−1}F ⋯ \frac{1}{i_{m} + j_{m} − 1}∂_{1}∂_{i_{m}+j_{m}−1}F ),

that is

∂_{i}∂_{j}F = ∑_{m=1}^{∞} ∑_{s_{1}+···+s_{m}=i+j} \frac{(−1)^{m+1}}{m} \frac{ij}{(s_{1} − 1) ⋯ (s_{m} − 1)}·

·P_{ij}(s_{1} − 1, ..., s_{m} − 1)∂_{1}∂_{s_{1}−1}F ⋯ ∂_{1}∂_{s_{m}−1}F.□

Remark. The equations

∂_{i}∂_{j}F = ∑_{m=1}^{∞} ∑_{s_{1}+···+s_{m}=i+j} \frac{(−1)^{m+1}}{m} \frac{ij}{(s_{1} − 1) ⋯ (s_{m} − 1)}·

·P_{ij}(s_{1} − 1, ..., s_{m} − 1)∂_{1}∂_{s_{1}−1}F ⋯ ∂_{1}∂_{s_{m}−1}F

describe the dispersionless limit of KP equation. Some other description of this hierarchy is presented in [ 11 ]. A comparison of these descriptions gives some nontrivial combinatorial identity P_{ij}.

Lemma 3.3. The following relation holds

∂_{i}∂_{j}F = ∑_{m=1}^{∞} ∑_{p_{1}+···+p_{m}=j+i} \frac{ij}{p_{1} ⋯ p_{m}} T_{ij}(p_{1} ⋯ p_{m})∂_{0}∂_{p_{1}}F ⋯ ∂_{0}∂_{p_{m}}F,

where

T_{ij}(p_{1}...p_{m}) = ∑ ; \begin{matrix}n_{1} + ⋯ + n_{k} = m \\ n_{i} > 0 \end{matrix} \frac{(−1)^{m+1}}{k} \frac{1}{n_{1}!···n_{k}!}P_{ij}(p_{1} + ⋯ + p_{q_{1}} − 1, ..., p_{q_{k−1}+1} + ⋯ +

p_{q_{k}} − 1), and q_{j} = ∑_{i=1}^{j} n_{i}.

Proof: According to lemmas 3.1 and 3.2,

∂_{i}∂_{j}F = ∑_{m=1}^{∞} ∑_{s_{1}+···+s_{m}=j+i} \frac{(−1)^{m+1}}{m} \frac{ij}{(s_{1} − 1) ⋯ (s_{m} − 1)}P_{ij}(s_{1} − 1, ⋯, s_{m} − 1)·

·∂_{1}∂_{s_{1}−1}F ⋯ ∂_{1}∂_{s_{m}−1}F) = ∑_{m=1}^{∞} ∑_{s_{1}+···+s_{m}=j+i} \frac{(−1)^{m+1}}{m} \frac{ij}{(s_{1} − 1) ⋯ (s_{m} − 1)}·

·P_{ij}(s_{1} − 1, ⋯, s_{m} − 1)( ∑_{n_{1}=1}^{∞} ∑_{p_{1}+···+p_{n_{1}} =s_{1}} \frac{(−1)^{n_{1}+1}}{n_{1}!} \frac{s_{1} − 1}{p_{1} ⋯ p_{n_{1}}} ∂_{0}∂_{p_{1}}F ⋯ ∂_{0}∂_{p_{n_{1}}} F )⋯

… ( ∑_{n_{m}=1}^{∞} ∑_{p_{1}+···+p_{nm} =s_{m}} \frac{(−1)^{n_{m}+1}}{n_{m}!} \frac{s_{m} − 1}{p_{1} ⋯ p_{n_{m}}} ∂_{0}∂_{p_{1}}F ⋯ ∂_{0}∂_{p_{nm}} F )=

= ∑_{m=1}^{∞} ∑_{p_{1}+···+p_{m}=j+i} \frac{ij}{p_{1} ⋯ p_{m}} T_{ij}(p_{1} ⋯ p_{m})∂_{0}∂_{p_{1}}F ⋯ ∂_{0}∂_{p_{m}}F.□

Using the induction, we get from lemma 3.3 Lemma 3.4. The following relation holds

∂_{i_{1}}∂_{i_{2}} ⋯ ∂_{i_{k}}F = ∑_{m=1}^{∞}( ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = i_{1} + ⋯ + i_{k} \\ ℓ_{1} + ⋯ + ℓ_{m} = m + k − 2 \\ s_{j}, ℓ_{j} ⩾ 1 \end{matrix} \frac{i_{1} ⋯ i_{k}}{s_{1} ⋯ s_{m}} ·

·T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ∂_{0}^{ℓ_{1}}∂_{s_{1}}F ⋯ ∂_{0}^{ℓ_{m}}∂_{s_{m}}F ),

where T_{i_{1},i_{2}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) = \{ \begin{matrix}T_{i_{1}i_{2}}(s_{1} ⋯ s_{m}), & if ℓ_{1} = ⋯ = ℓ_{m} = 1 \\ 0 & in another cases \end{matrix} ,

T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) = ∑ ; \begin{matrix}1 ⩽ i ⩽ j ⩽ m \\ s, ℓ > 0 \end{matrix} T_{i_{1}···i_{k−1}}( \begin{matrix}s_{1} ⋯ s_{i−1} \\ ℓ_{1} ⋯ ℓ_{i−1} \end{matrix} ( \begin{matrix}s \\ ℓ \end{matrix} ) \begin{matrix}s_{j+1} ⋯ s_{m} \\ ℓ_{j+1} ⋯ ℓ_{m} \end{matrix} )·

·T_{s,i_{k}}(s_{i}, s_{i+1}, ..., s_{j})\frac{ℓ!}{(ℓ_{i} − 1)! ⋯ (ℓ_{j} − 1)!},

and s = s_{i} + s_{i+1} + ⋯ + s_{j} − i_{k}, ℓ = (ℓ_{i} − 1) + ⋯ + (ℓ_{j} − 1).□

Define now the Cauchy data for F. Lemma 3.5. The following relations hold

∂_{0}F |_{t_{0}} = −t_{0} + t_{0} ln t_{0} + const and ∂_{k}F |_{t_{0}} = 0 for k > 0.

Proof: If t_{1} = t_{2} = ⋯ = 0, then Q_{∘}^{t} = \{w ∈ ℂ||w| < t0^{\frac{1}{2}} \} and thus, w(z, t)|_{t_{0}} = Rz = t0^{\frac{1}{2}} z.

It is follow from theorem 2.1 that

∂_{0}F |_{t_{0}}= \frac{2}{π} ∫_{|z|⩽R} ln |z|dxdy = \frac{2}{π} ∫_{0}^{R} dr ∫_{0}^{2π}dφ · r · ln |r| =

= \frac{2 · 2π}{2π} ∫_{0}^{R} ln |r|dr^{2} = 2(r^{2} ln r|_{0}^{R}− ∫_{0}^{R} rdr) =

= r^{2} ln r^{2}|_{0}^{R}−2\frac{1}{2}r^{2}|_{0}^{R}= R^{2} ln R^{2} − R^{2} = −t_{0} + t_{0} ln t_{0}.

∂_{k}F |_{t_{0}} = \frac{1}{π} ∫_{Q_{∘}^{t}} z^{k}dxdy = 0 for k > 0. □

Lemma 3.6. The following relations hold

∂_{i} ¯∂_{j}F |_{t_{0}} = \{ \begin{matrix}0 & for i ≠ j, \\ it_{0}^{i} & for i = j. \end{matrix}

Proof: It is follow from lemma 3.5 that ∂_{0}∂_{k}F |_{t_{0}}= 0 for k > 0 and ∂_{0}^{2}F |_{t_{0}} = ln t_{0}. Thus,

e^{∂_{0}(∂_{0}+D(z)+D̄(ξ̄))F} |_{t_{0}}= t_{0}.

Moreover, according to (1.5(c)),

1 − e^{−D(z)D̄(ξ̄)F} = z^{−1}ξ̄^{−1}e^{∂_{0}(∂_{0}+D(z)+D̄(ξ̄))F} ,

and thus,

−D(z)D̄(ξ̄)F |_{t_{0}}= ln(1 − z^{−1}ξ̄^{−1}t_{0}) = − ∑_{k=1}^{∞} kz^{−k}ξ^{−k}t_{0}^{k}.

Therefore ∂_{i} ¯∂_{j}F |_{t_{0}}= 0 for i ≠ j and ∂_{i} ¯∂_{i}F |_{t_{0}}= it_{0}^{i} □.

Lemma 3.7. The following relations hold

∂_{i} ¯∂_{i_{1}} ⋯ ¯∂_{i_{k}}F |_{t_{0}}= ¯∂_{i}∂_{i_{1}} ⋯ ∂_{i_{k}}F |_{t_{0}}= \{ \begin{matrix}0, & if i_{1} + ⋯ + i_{k} ≠ i \\ i_{1} ⋯ i_{k} \frac{i!}{(i−k+1)!}t_{0}^{i−k+1}, & if i = i_{1} + ⋯ + i_{k} \end{matrix} .

Proof: The differentials ∂ and ¯∂ occur in (1.5) symmetrically. This gives the first equality. Moreover, according to lemmas 3.3 and 3.4,

∂_{i_{1}}∂_{i_{2}} ⋯ ∂_{i_{k}}F = \frac{i_{1} ⋯ i_{k}}{i} T_{i_{1}···i_{k}}( \begin{matrix}i \\ k − 1 \end{matrix} )∂_{0}^{k−1}∂_{i}F +

+ ∑_{m=2}^{∞}( ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = i_{1} + ⋯ + i_{k} \\ ℓ_{1} + ⋯ ℓ_{m} = m + k − 2 \\ s_{j}, t_{j} ⩾ 1 \end{matrix} \frac{i_{1} ⋯ i_{k}}{s_{1} ⋯ s_{m}} T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ·

·∂_{0}^{ℓ_{1}}∂_{s_{1}}F ⋯ ∂_{0}^{ℓ_{m}}∂_{s_{m}}F ) = \frac{i_{1} ⋯ i_{k}}{i} ∂_{0}^{k−1}∂_{i}F +

+ ∑_{m=2}^{∞}( ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = i_{1} + ⋯ + i_{k} \\ ℓ_{1} + ⋯ ℓ_{m} = m + k − 2 \\ s_{j}, t_{j} ⩾ 1 \end{matrix} \frac{i_{1} ⋯ i_{k}}{s_{1} ⋯ s_{m}} T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ·

·∂_{0}^{ℓ_{1}}∂_{s_{1}}F ⋯ ∂_{0}^{ℓ_{m}}∂_{s_{m}}F ),

where i = i_{1} + ⋯ + i_{k}. This equality and lemmas 3.5 and 3.6 give the second equality in the affirmation of lemma 3.7. □

Lemma 3.8. The following relation holds

∂_{i_{1}} ⋯ ∂_{i_{k}} ¯∂ī_{1} ⋯ ¯∂īk̄F |_{t_{0}}= ∑_{i=1}^{∞} Ñ_{i}(i_{1} ⋯ i_{k}|ī_{1} ⋯ īk̄)t_{0}^{i−(k+k̄)+2}, where Ñ_{i}(i_{1} ⋯ i_{k}|ī_{1} ⋯ īk̄) = 0

if i − (k + k̄) + 2 < 0, ∑_{j=1}^{k} i_{j} ≠ i or ∑_{j=1}^{k̄} ī_{j} ≠ i. In opposite case, Ñ_{i}(i_{1} ⋯ i_{i_{k}} |ī_{1}, ⋯ īk̄) =

= ∑_{m=1}^{∞}( ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = i_{1} + ⋯ + i_{k} \\ ℓ_{1} + ⋯ ℓ_{m} = m + k − 2 \\ s_{j}, ℓ_{j} ⩾ 1 \end{matrix} i_{1} ⋯ i_{k}ī_{k} ⋯ īk̄·

·T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) Sī_{1}_{···}īk̄ ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} )),

where

Sī_{1}_{···}īk̄ ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) = ∑ ; \begin{matrix}n_{1} + ⋯ + n_{m} = k \\ s_{i} − n_{i} + 1 − ℓ_{i} ⩾ 0 \end{matrix} ∑ \frac{(s_{1} − 1)! ⋯ (s_{m} − 1)!}{(s_{1} − N_{1} + 1 − ℓ_{1})! ⋯ (s_{m} − n_{m} + 1 − ℓ_{m})!}

and second sum is the sum by all partitions of the set {ī_{1}, ..., īk̄} on subsets

{j_{1}^{p}, ..., j_{n_{p}}^{p}} (p = 1, ..., m) such that ∑_{α=1}^{n_{p}} j_{α}^{p} = s_{p}. □

Proof: According to lemma 3.4,

∂_{i_{1}} ⋯ ∂_{i_{k}} ¯∂ī_{1} ⋯ ¯∂īk̄F = ¯∂īī ⋯ ¯∂īk̄( ∑_{m=1}^{∞}( ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = i_{1} + ⋯ + i_{k} \\ ℓ_{1} + ⋯ + ℓ_{m} = m + k − 2 \\ s_{j}, ℓ_{j} ⩾ 1 \end{matrix} \frac{i_{1} ⋯ i_{k}}{s_{1} ⋯ s_{m}} ·

· T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ∂_{0}^{ℓ_{1}}∂_{s_{1}}F ⋯ ∂_{0}^{ℓ_{m}}∂_{s_{m}}F ))=

= ∑( ∑_{m=1}^{∞}( ∑ ; \begin{matrix}s_{1} + ⋯ + s_{m} = i_{1} + ⋯ + i_{k} \\ ℓ_{1} + ⋯ + ℓ_{m} = m + k − 2 \\ s_{j}, ℓ_{j} ⩾ 1 \end{matrix} \frac{i_{1} ⋯ i_{k}}{s_{1} ⋯ s_{m}} T_{i_{1}···i_{k}} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ·

· ∂_{0}^{ℓ_{1}}∂_{s_{1}} ¯∂1 ; j̄_{1} ⋯ ¯∂1 ; j̄_{n_{1}} F ⋯ ∂_{0}^{ℓ_{m}}∂_{s_{m}} ¯∂m ; j̄_{1} ⋯ ¯∂m ; j̄_{nm} F )),

where the first sum is taken by all partitions of {ī_{1}, ..., īk̄} on subset {(j_{i}^{1}, ..., j_{n_{1}}^{1})...(j_{1}^{m}, ..., j_{n_{m}}^{m})}. According to lemma 3.7, this gives the affirmation of lemma 3.8 □.

Lemma 3.8 gives

Lemma 3.9 The following relation holds

F = \frac{1}{2}t_{0}^{2} log t_{0} − \frac{3}{4}t_{0}^{2} + ∑ ; \begin{matrix}i_{1} < ⋯ < i_{k} \\ ī_{1} < ⋯ īk̄ \end{matrix} ∑_{n_{j},n̄_{j}=1}^{∞} \frac{1}{n_{1}! ⋯ n_{k}!n̄_{1}! ⋯ n̄k̄!}Ñ_{i}( \begin{matrix}i_{1}...i_{k} \\ n_{1}...n_{k} \end{matrix} | \begin{matrix}ī_{1}...īk̄ \\ n̄_{1}...n̄k̄ \end{matrix} )·

·t_{0}^{i−( ∑_{i=1}^{k} n_{i}+ ∑_{i=1}^{k̄} n̄_{i})+2}t_{i_{1}}^{n_{1}} ⋯ t_{i_{k}}^{n_{k}} t̄n̄_{1} ; ī_{1} ⋯ t̄n̄k̄ ; īk̄,

where

Ñ_{i}( \begin{matrix}i_{1}...i_{k} \\ n_{1}...n_{k} \end{matrix} | \begin{matrix}ī_{1}...īk̄ \\ n̄_{1}...n̄k̄ \end{matrix} ) = 0,

if i − (∑_{i=1}^{k} n_{i} + ∑_{i=1}^{k̄} n̄_{i}) + 2 < 0, i ≠ ∑_{j=1}^{k} n_{j}i_{j} or i ≠ ∑_{j=1}^{k̄} n̄_{j}ī_{j}. In opposite case

Ñ_{i}( \begin{matrix}i_{1}...i_{k} \\ n_{1}...n_{k} \end{matrix} | \begin{matrix}ī_{1}...īk̄ \\ |n_{1}...n̄k̄ \end{matrix} ) = Ñ_{i}(i_{1}...i_{1}i_{2}...i_{2}...i_{k}...i_{k}|ī_{1}...ī_{1}ī_{2}...ī_{2}...īk̄...īk̄),

where in the last parentheses any i_{j} (respectively any ī_{j} ) occurs n_{j} (respectively n̄_{j}) cases. The theorem 3.1 is equivalent to lemma 3.9. Using the algorithm of the present paper, Yu.Klimov and A.Korzh created a computer

program for calculations of the coefficients of the Taylor series F.

4. Convergence conditions of the Taylor series for the potential.

The combinatorial coefficients N_{i}^{2}(...) have some remarkable properties. For example,

Theorem 4.1. N_{i}^{2} ( \begin{matrix}i_{1} ⋯ i_{k} \\ n_{1} ⋯ n_{k} \end{matrix} | 1 ; n̄_{1} ) = \{ \begin{matrix}(i − 1)!, if k = n_{1} = 1, i = i_{1} = n̄_{1}, \\ 0 in the other cases. \end{matrix}

Proof. According to our definition

S\underbrace{1, …, 1}_{k̄} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) =

= ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{\underbrace{1, …, 1}\} \\ ī_{r}^{1}+···+ī_{r}^{nr} =s_{r} \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1} − l_{1} + 1)!(l_{1} − 1)!} × ⋯ ×

× ⋯ × \frac{(s_{m} − 1)!}{(s_{m} − n_{m} − l_{m} + 1)!(l_{m} − 1)!} = δ_{ℓ_{1},1} … δ_{ℓ_{m},1} \frac{k̄!}{s_{1} … s_{m}} .

Thus, if k > 2, then

N_{i}^{1}(i_{1}, …, i_{k}|1, …, 1) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+···+i_{k} \\ l_{1}+···+l_{m}=m+k−2 \\ s_{r},l_{r}⩾1 \end{matrix}}(−1)^{m+1} S_{1,...,1} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) ×

× T_{i_{1},...,i_{k}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) = 0 .

Let now k = 2 (i_{1}, i_{2} ⩾ 1). Then

N_{i}^{1}(i_{1}, i_{2}| \underbrace{1, …, 1}_{k̄}) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+i_{2} \\ s_{r}⩾1 \end{matrix}}(−1)^{m+1} S\underbrace{1, …, 1}_{k̄} ( \begin{matrix}s_{1}, & …, & s_{m} \\ 1, & …, & 1 \end{matrix} ) ×

× T_{i_{1},i_{2}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ 1, & …, & 1 \end{matrix} ) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+i_{2} \\ s_{r}⩾1 \end{matrix}}(−1)^{m+1} \frac{k̄!}{s_{1} … s_{m}} T_{i_{1},i_{2}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ 1, & …, & 1 \end{matrix} ) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+i_{2} \\ s_{r}⩾1 \end{matrix}}(−1)^{m+1} \frac{k̄!}{s_{1} … s_{m}} T_{i_{1},i_{2}}^{1}(s_{1}, …, s_{m}) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+i_{2} \\ s_{r}⩾1 \end{matrix}}(−1)^{m+1} \frac{k̄!}{s_{1} … s_{m}} ∑_{\begin{matrix}k⩾1 \\ n_{1}+···+n_{k}=m \\ n_{r}⩾1 \end{matrix}} \frac{1}{kn_{1}! … n_{k}!}×

× P_{i,j} (\underbrace{s_{1} + ⋯ + s_{n_{1}}}_{n_{1}}, …, \underbrace{s_{n_{1}+···+n_{k−1}+1} + ⋯ + s_{n_{1}+···+n_{k}}}_{n_{k}}) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+i_{2} \\ k⩾1 \\ n_{1}+···+n_{k}=m \end{matrix}} \frac{(−1)^{m+1}}{kn_{1}! … n_{k}!} \frac{k̄!}{s_{1} … s_{m}} ×

× P_{i,j} (\underbrace{s_{1} + ⋯ + s_{n_{1}}}_{n_{1}}, …, \underbrace{s_{n_{1}+···+n_{k−1}+1} + ⋯ + s_{n_{1}+···+n_{k}}}_{n_{k}}) =

= ∑_{\begin{matrix}k⩾1 \\ s̃_{1}+···+s̃_{k}=i_{1}+i_{2} \end{matrix}} P_{i,j}(s̃_{1}, …, s̃_{k})×

× ∑_{\begin{matrix}n_{r}⩾1, n_{1}+···+n_{k}=m \\ \underbrace{s_{n_{1}+···+n_{r−1}+1} + ⋯ + s_{n_{1}+···+n_{r}}}_{nr}=s̃_{r} \end{matrix}} \frac{(−1)^{m+1}}{kn_{1}! … n_{k}!} \frac{k̄!}{s_{1} … s_{m}} =

= ∑_{\begin{matrix}k⩾1 \\ s̃_{1}+···+s̃_{k}=i_{1}+i_{2} \end{matrix}} −\frac{k̄!}{k} P_{i,j}(s̃_{1}, …, s̃_{k})×

× ∏_{1⩽r⩽k} ∑_{\begin{matrix}n_{r}⩾1 \\ s_{1}+···+s_{nr} =s̃_{r} \end{matrix}} \frac{(−1)^{n_{r}}}{n_{r}!s_{1} … s_{n_{r}}} .

In addition if s > 1, then

∑_{\begin{matrix}n⩾1 \\ s_{1}+···+s_{n}=s \end{matrix}} \frac{(−1)^{n}}{n!s_{1} … s_{n}} = \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} ∑_{n⩾1} \frac{(−1)^{n}}{n!} (x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + …)^{n}|_{x=0} =

= \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} ∑_{n⩾1} \frac{(−1)^{n}}{n!} (− log(1 − x))^{n}|_{x=0} =

= \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} ∑_{n⩾1} \frac{(log(1 − x))^{n}}{n!} |_{x−0} = \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} (exp(log(1 − x)) − 1)|_{x=0} = 0.

Thus,

N_{i}^{1}(i_{1}, i_{2}| \underbrace{1, …, 1}_{k̄}) = ∑_{\begin{matrix}k⩾1 \\ s̃_{1}+···+s̃_{k}=i_{1}+i_{2} \end{matrix}} −\frac{k̄!}{k} P_{i,j}(s̃_{1}, …, s̃_{k})×

× ∏_{1⩽r⩽k} ∑_{\begin{matrix}n_{r}⩾1 \\ s_{1}+···+s_{nr} =s̃_{r} \end{matrix}} \frac{(−1)^{n_{r}}}{n_{r}!s_{1} … s_{n_{r}}} = 0 ,

because P_{i,j}(1, …, 1) = 0.

If k = 1, then from our definition it follows that

N_{i}^{2} ( \begin{matrix}i_{1} \\ n_{1} \end{matrix} | 1 ; n̄_{1} ) = \{ \begin{matrix}(i − 1)!, if n_{1} = 1, i = i_{1} = n̄_{1}, \\ 0 in the other cases. \end{matrix} . □

If t_{i}, t̄_{i} = 0 for i > 2 , then ∂Q is ellipse. In this case theorem 3.1 gives

F = −\frac{3}{4}t_{0}^{2} + \frac{1}{2}t_{0}^{2} log (\frac{t_{0}}{1 − 4|t_{2}|^{2}} ) + \frac{t_{0}}{1 − 4|t_{2}|^{2}} (|t_{1}|^{2} + t_{1}^{2}t̄_{2} + t̄_{1}^{2}t_{2}).

This formula was at first obtained in [ 3 ], using formulas for conforming maps from ellipse to circle.

The recurrent formulas for coefficients of the Taylor series F give possibility to estimate the coefficients and to find sufficient convergence conditions for F, if t_{i}, t̄_{i} = 0 for i > n.

16 S.NATANZON

Theorem 4.2. Let t̃ = (t_{0}, t_{1}, t̄_{1}, t_{2}, t̄_{2}, ...) be such that t_{i}, t̄_{i} = 0 for i > n, 0 < t_{0} < 1 and |t_{i}|, |t̄_{i}| ⩽ (4n^{3}2^{n}e^{n})^{−1}. Then the series F (t̃) is convergence.

A proof is based on a sequential of estimations of all values, utilised in the definition of N^{2}. Present these estimations: Lemma 4.1. Let i + j = s_{1} + ⋯ + s_{m}. Then P_{ij}(s_{1}..., s_{m}) ⩽ min(C_{i−1}^{m−1}, C_{j−1}^{m−1}). Proof.

P_{j,i}(s_{1}, …, s_{m}) = P_{i,j}(s_{1}, …, s_{m}) = #{(i_{1}, …, i_{m}) | i =
= i_{1} + ⋯ + i_{m}, 1 ⩽ i_{r} ⩽ s_{r} − 1} ⩽
⩽ #{(i_{1}, …, i_{m})|i = i_{1} + ⋯ + i_{m}, 1 ⩽ i_{r}} = C_{i−1}^{m−1} . □

Lemma 4.2. Let i + j = s_{1} + ⋯ + s_{m}. Then T_{ij}^{1}(s_{1}, ..., s_{m}) ⩽ \frac{ℓ^{m−1}}{m!} , where ℓ = min(i, j). Proof.

T_{j,i}^{1}(s_{1}, …, s_{m}) = T_{i,j}^{1}(s_{1}, …, s_{m}) = ∑_{\begin{matrix}k⩾1 \\ n_{1}+···+n_{k}=m \\ n_{r}⩾1 \end{matrix}} \frac{1}{kn_{1}! … n_{k}!} ×

× P_{i,j} (\underbrace{s_{1} + ⋯ + s_{n_{1}}}_{n_{1}}, …, \underbrace{s_{n_{1}+···+n_{k−1}+1} + ⋯ + s_{n_{1}+···+n_{k}}}_{n_{k}}) ⩽

⩽ ∑_{\begin{matrix}k⩾1 \\ n_{1}+···+n_{k}=m \\ n_{r}⩾1 \end{matrix}} \frac{C_{i−1}^{k−1}}{kn_{1}! … n_{k}!} =

= \frac{1}{m!} \frac{∂^{m}}{∂x^{m}} \frac{1}{i} ∑_{k⩾1} C_{i}^{k} ∑_{n_{1}+···+n_{k}=m} \frac{1}{n_{1}! … n_{k}!} =

\frac{1}{m!} \frac{∂^{m}}{∂x^{m}} \frac{1}{i} ∑_{k⩾1} C_{i}^{k} (x + \frac{x^{2}}{2!} + …)^{k}|_{x=0} =

= \frac{1}{m!} \frac{∂^{m}}{∂x^{m}} \frac{1}{i} (∑_{k⩾1} C_{i}^{k}(e^{x} − 1)^{k})|_{x=0} =

= \frac{1}{m!} \frac{∂^{m}}{∂x^{m}} \frac{1}{i} ((1 + (e^{x} − 1))^{i} − 1)|_{x=0} = \frac{1}{m!} \frac{∂^{m}}{∂x^{m}} \frac{1}{i} (e^{ix} − 1)|_{x=0} = \frac{i^{m−1}}{m!} . □

TOWARDS AN EFFECTIVISATION OF THE RIEMANN THEOREM 17

Lemma 4.3. Let i_{1} + ⋯ + i_{k} = s_{1} + ⋯ + s_{m} and (ℓ_{1} − 1) + ⋯ + (ℓ_{m} − 1) = k − 2. Then

T_{i_{1}...,i_{k}}^{2} ( \begin{matrix}s_{1} ⋯ s_{m} \\ ℓ_{1} ⋯ ℓ_{m} \end{matrix} ) ⩽ \frac{I^{m−1}(k−1)^{m}(k−2)!}{m!} , where I = max(i_{r}).

Proof. We use an induction by k. If k = 2, then

T_{i_{1},i_{2}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ 1, & …, & 1 \end{matrix} ) = T_{i_{1},i_{2}}^{1}(s_{1}, …, s_{m}) ⩽

⩽ \frac{I^{m−1}}{m!} = \frac{I^{m−1}(k − 1)^{m}(k − 2)!}{m!}|_{k = 2} .

Let k > 2. Note at first that, if l = (l_{i} − 1) + ⋯ + (l_{j} − 1), then

∑_{1⩽i⩽j⩽m, j−i=d} \frac{l}{(k − 2)(d + 1)} = \frac{∑_{1⩽i⩽j⩽m, j−i=d}(l_{i} − 1) + ⋯ + (l_{j} − 1)}{(k − 2)(d + 1)} ⩽

⩽ \frac{∑_{1⩽t⩽m}(l_{t} − 1)(d + 1)}{(k − 2)(d + 1)} = 1.

Thus,

T_{i_{1},...,i_{k}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) = ∑_{\begin{matrix}1⩽i⩽j⩽m \\ s, l⩾1 \end{matrix}} l T_{s,i_{k}}^{1}(s_{i}, …, s_{j}) ×

× T_{i_{1},...,i_{k−1}}^{2} ( \begin{matrix}s_{1}, & …, & s_{i−1}, & s, & s_{j+1}, & …, & s_{m} \\ l_{1}, & …, & l_{i−1}, & l, & l_{j+1}, & …, & l_{m} \end{matrix} ) ⩽

⩽ ∑_{1⩽i⩽j⩽m} l \frac{I^{j−i}}{(j − i + 1)!} \frac{I^{m−(j−i+1)+1−1}(k − 2)^{m−(j−i+1)+1}(k − 3)!}{(m − (j − i + 1) + 1)!} =

= I^{m−1}(k − 3)! ∑_{1⩽i⩽j⩽m} l \frac{(k − 2)^{m−(j−i)}}{(j − i + 1)!(m − (j − i))!} =

= \frac{I^{m−1}(k − 2)!}{m!} ∑_{1⩽i⩽j⩽m, d=j−i} \frac{l}{(k − 2)(d + 1)} \frac{m!}{d!(m − d)!}(k − 2)^{m−d} =

= \frac{I^{m−1}(k − 2)!}{m!} ∑_{d=0}^{m−1} \frac{m!}{d!(m − d)!}(k − 2)^{m−d} ( ∑_{1⩽i⩽j⩽m, j−i=d} \frac{l}{(k − 2)(d + 1)}) ⩽

⩽ \frac{I^{m−1}(k − 2)!}{m!} ∑_{d=0}^{m−1} \frac{m!}{d!(m − d)!}(k − 2)^{m−d} ⩽ \frac{I^{m−1}(k − 2)!}{m!} ∑_{t=0,t=m−d}^{m} C_{m}^{t}(k − 2)^{t} =

= \frac{I^{m−1}(k − 2)!}{m!}((k − 2) + 1)^{m} = \frac{I^{m−1}(k − 1)^{m}(k − 2)!}{m!} . □

Lemma 4.4. Let Ī = max(ī_{r}) and

S̃_{ī_{1},...,īk̄}(m, k) = ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ (l_{1}−1)+···+(l_{m}−1)=k−2 \\ n_{r}, l_{r}⩾1 \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1} − l_{1} + 1)!(l_{1} − 1)!} × ⋯ ×

× \frac{(s_{m} − 1)!}{(s_{m} − n_{m} − l_{m} + 1)!(l_{m} − 1)!} .

Then S̃_{ī_{1},...,īk̄}(m, k) ⩽ m(k̄ − 1)!C_{Īk̄−k̄}^{k−2}C_{Īk̄}^{k̄−m}. Proof. We use the equality

∑_{ñ_{1}+···+ñ_{m}=k̄−m} Cñ_{1} ; Īñ_{1}+Ī × ⋯ × Cñ_{m} ; Īñ_{m}+Ī\frac{Ī}{Īñ_{1} + Ī} × ⋯ × \frac{Ī}{Īñ_{m} + Ī} =

= C_{Ī(k̄−m)+mĪ}^{k̄−m}\frac{mĪ}{Ī(k̄ − m) + mĪ}

that follows from [ 12 ]. Then

S̃_{ī_{1},...,īk̄}(m, k) =

= ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ (l_{1}−1)+···+(l_{m}−1)=k−2 \\ n_{r}, l_{r}⩾1 \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1} − l_{1} + 1)!(l_{1} − 1)!} × ⋯ ×

×\frac{(s_{m} − 1)!}{(s_{m} − n_{m} − l_{m} + 1)!(l_{m} − 1)!} =

= ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} ⩾n_{r}⩾1 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1})!} × ⋯ × \frac{(s_{m} − 1)!}{(s_{m} − n_{m})!}×

× ∑_{\begin{matrix}(l_{1}−1)+···+(l_{m}−1)=k−2 \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1} − n_{1})!}{(s_{1} − n_{1} − l_{1} + 1)!(l_{1} − 1)!} × ⋯ ×

TOWARDS AN EFFECTIVISATION OF THE RIEMANN THEOREM 19

× \frac{(s_{m} − n_{m})!}{(s_{m} − n_{m} − l_{m} + 1)!(l_{m} − 1)!} =

= ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} ⩾n_{r}⩾1 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1})!} × ⋯ ×

×\frac{(s_{m} − 1)!}{(s_{m} − n_{m})!} ∑_{l̃_{1}+···+l̃_{m}=k−2} Cl̃_{1} ; s_{1}−n_{1} … Cl̃_{m} ; s_{m}−n_{m} =

= ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} ⩾n_{r}⩾1 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1})!} × ⋯ ×

× \frac{(s_{m} − 1)!}{(s_{m} − n_{m})!}C_{(s_{1}+···+s_{m})−(n_{1}+···+n_{m})}^{k−2} =

= C_{ī_{1}+···+īk̄−k̄}^{k−2} ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} ⩾n_{r}⩾1 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1})!} × ⋯ × \frac{(s_{m} − 1)!}{(s_{m} − n_{m})!} ⩽

⩽ S̃\underbrace{Ī, …, Ī}_{k̄}(m, k) = C_{Īk̄−k̄}^{k−2} ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔ \\ ⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{Ī,...,Ī\}, n_{r}⩾1 \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} =In_{r} \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1})!} × ⋯ × \frac{(s_{m} − 1)!}{(s_{m} − n_{m})!} =

= C_{Īk̄−k̄}^{k−2} ∑_{\begin{matrix}n_{1}+···+n_{m}=k̄ \\ n_{r}⩾1 \end{matrix}} \frac{k̄!}{n_{1}! … n_{m}!} \frac{(Īn_{1} − 1)!}{(Īn_{1} − n_{1})!} × ⋯ × \frac{(Īn_{m} − 1)!}{(Īn_{m} − n_{m})!} =

= k̄!C_{Īk̄−k̄}^{k−2} ∑_{\begin{matrix}ñ_{1}+···+ñ_{m}=k̄−m \\ ñ_{r}=n_{r}−1⩾0 \end{matrix}} \frac{(Īñ_{1} + Ī− 1)!}{(ñ_{1} + 1)!(Īñ_{1} + Ī− ñ_{1} − 1)!} × ⋯ ×

× \frac{(Īñ_{m} + Ī− 1)!}{(ñ_{m} + 1)!(Īñ_{m} + Ī− ñ_{m} − 1)!} =

= k̄!C_{Īk̄−k̄}^{k−2} ∑_{ñ_{1}+···+ñ_{m}=k̄−m} \frac{Īñ_{1} + Ī− ñ_{1}}{(Īñ_{1} + Ī)(ñ_{1} + 1)} \frac{(Īñ_{1} + Ī)!}{ñ_{1}!(Īñ_{1} + Ī− ñ_{1})!} × ⋯ ×

× \frac{Īñ_{m} + Ī− ñ_{m}}{(Īñ_{m} + Ī)(ñ_{m} + 1)} \frac{(Īñ_{m} + Ī)!}{ñ_{m}!(Īñ_{m} + Ī− ñ_{m})!} ⩽

⩽ k̄!C_{Īk̄−k̄}^{k−2} ∑_{ñ_{1}+···+ñ_{m}=k̄−m} \frac{Ī}{Īñ_{1} + Ī}Cñ_{1} ; Īñ_{1}+Ī × ⋯ × \frac{Ī}{Īñ_{m} + Ī}Cñ_{m} ; Īñ_{m}+Ī =

= k̄!C_{Īk̄−k̄}^{k−2}C_{Ī(k̄−m)+mĪ}^{k̄−m}\frac{mĪ}{Ī(k̄ − m) + mĪ} ⩽ m(k̄ − 1)!C_{Īk̄−k̄}^{k−2}C_{Īk̄}^{k̄−m} . □

Lemma 4.5. N_{i}^{1}(i_{1}..., i_{k}|ī_{1}..., īk̄) ⩽ (k − 1)!(k̄ − 1)!e^{I(k−1)}2^{Īk̄−k̄}2^{Īk̄}. Proof.

N_{i}^{1}(i_{1}, …, i_{k}|ī_{1}, …, īk̄) =

= ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+···+i_{k} \\ l_{1}+···+l_{m}=m+k−2 \\ s_{r},l_{r}⩾1 \end{matrix}}(−1)^{m+1} S_{ī_{1},...,īk̄} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) ×

× T_{i_{1},...,i_{k}}^{2} ( \begin{matrix}s_{1}, & …, & s_{m} \\ l_{1}, & …, & l_{m} \end{matrix} ) ⩽

⩽ ∑_{\begin{matrix}m⩾1 \\ s_{1}+···+s_{m}=i_{1}+···+i_{k} \\ l_{1}+···+l_{m}=m+k−2 \\ s_{r},l_{r}⩾1 \end{matrix}} \frac{I^{m−1}(k − 1)^{m}(k − 2)!}{m!}×

× ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ ī_{r}^{1}+···+ī_{r}^{nr} =s_{r} \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1} − l_{1} + 1)!(l_{1} − 1)!} × ⋯ ×

× \frac{(s_{m} − 1)!}{(s_{m} − n_{m} − l_{m} + 1)!(l_{m} − 1)!} =

= ∑_{m⩾1} \frac{I^{m−1}(k − 1)^{m−1}(k − 2)!}{m!}×

× ∑_{\begin{matrix}\{ī_{1}^{1},...,ī_{1}^{n_{1}} \}⊔···⊔\{ī_{m}^{1},...,ī_{m}^{nm} \}= \\ =\{ī_{1},...,īk̄\} \\ (l_{1}−1)+···+(l_{m}−1)=k−2 \\ n_{r}, l_{r}⩾1 \\ s_{r}=ī_{r}^{1}+···+ī_{r}^{nr} \\ s_{r}−n_{r}−ℓ_{r}+1⩾0 \end{matrix}} \frac{(s_{1} − 1)!}{(s_{1} − n_{1} − l_{1} + 1)!(l_{1} − 1)!} × ⋯ ×

×\frac{(s_{m} − 1)!}{(s_{m} − n_{m} − l_{m} + 1)!(l_{m} − 1)!} =

= ∑_{m⩾1} \frac{I^{m−1}(k − 1)^{m−1}(k − 2)!}{m!} S̃_{ī_{1},...,īk̄}(m, k) ⩽

⩽ ∑_{m⩾1} \frac{I^{m−1}(k − 1)^{m}(k − 2)!}{m!} S̃_{ī_{1},...,īk̄}(m, k) ⩽

⩽ ∑_{m⩾1} \frac{I^{m−1}(k − 1)^{m}(k − 2)!}{m!}m(k̄ − 1)!C_{Īk̄−k̄}^{k−2}C_{Īk̄}^{k̄−m} =

= (k − 1)!(k̄ − 1)! ∑_{m⩾1} \frac{I^{m−1}(k − 1)^{m−1}}{(m − 1)!} C_{Īk̄−k̄}^{k−2} C_{Īk̄}^{k̄−m} ⩽

⩽ (k − 1)!(k̄ − 1)!e^{I(k−1)}2^{Īk̄−k̄}2^{Īk̄} . □

Proof of theorem 4.2. The coefficient for t_{0}^{...}t_{i_{1}}^{n_{1}} … t_{i_{I}}^{n_{I}} t̄n̄_{1} ; ī_{1} … t̄n̄Ī ; īĪ is equal

\frac{i_{1}^{n_{1}} … i_{I}^{n_{I}}}{n_{1}! … n_{I}!} \frac{īn̄_{1} ; 1 … īn̄Ī ; Ī}{n̄_{1}! … n̄Ī!} N_{i}^{2} ( \begin{matrix}i_{1}, & …, & i_{I} \\ n_{1}, & …, & n_{I} \end{matrix} | \begin{matrix}ī_{1}, & …, & īĪ \\ n̄_{1}, & …, & n̄Ī \end{matrix} ) =

= \frac{i_{1}^{n_{1}} … i_{I}^{n_{I}}}{n_{1}! … n_{I}!} \frac{īn̄_{1} ; 1 … īn̄Ī ; Ī}{n̄_{1}! … n̄Ī!} N_{i}^{1} (\underbrace{i_{1}, …, i_{1}}_{n_{1}}, …, \underbrace{i_{I}, …, i_{I}}_{n_{I}} |\underbrace{ī_{1}, …, ī_{1}}_{n̄_{1}}, …,\underbrace{īk̄, …, īĪ}_{n̄Ī} ) ⩽

⩽ \frac{i_{1}^{n_{1}} … i_{I}^{n_{I}}}{n_{1}! … n_{I}!} \frac{īn̄_{1} ; 1 … īn̄Ī ; Ī}{n̄_{1}! … n̄Ī!}k!k̄!e^{Ĩ(k−1)}2^{2Ĩk̄} ⩽ Ĩ^{K}e^{ĨK}2^{ĨK} \frac{k!k̄!}{n_{1}! … n_{I}!n̄_{1}! … n̄Ī!} ⩽

⩽ Ĩ^{K}e^{ĨK}2^{ĨK} Ĩ^{K} ⩽ (Ĩ^{2}2ĨeĨ)^{K} ,

where k = n_{1} + ⋯ + n_{I}, k̄ = n̄_{1} + ⋯ + n̄_{I}, K = k + k̄ and Ĩ = max(I, Ī).

Consider now monomials from t_{0}, t_{1}, t̄_{1}, …, t_{n}, t̄_{n} of degree K. The number of such monomial is (2n)^{K}. Thus, its sum in the series is not more that

(n^{2}2^{n}e^{n})^{K}(2n)^{K}(4n^{3}2^{n}e^{n})^{−K} ⩽ 2^{−K} .

This implies the convergence of the series F (t̃). □

Reference.

1. P.S.Novikov, On uniqueness for a solution of the inverse problem for potetial. Soviet Math.Dok. 1938, v.18 N 3, 165-168.

2. A.Marshakov, A.Zabrodin, On the Dirichlet boundary problem and Hirota equations, e-print archiv: hep-th/0305259.

3. P.B.Wiegmann, A.Zabrodin, Conformal maps and dispersionless integrable hierarchies, Commun. Math.Phys. 213 (2000), 523.

4. K.Takasaki, T.Takebe, Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7(1995), 743-808.

5. R.Dijkgraaf, G.Moor, R.Plessner, The partition fanction of two-dimantional string theory, Nucl. Phys, B394 (1993), 356-382.

6. L.L.Chau, A.Zaboronsky, On the structure of correlation functions in the normal matrix model, Commun. Math. Thys., 196, 203 (1998).

7. S.Natanzon, Integrable systems and effectivisation of Riemann theorem about domains of the complex plane, e-print archiv: math.CV/0103136.

8. Yu.Klimov, A.Korzh, S.Natanzon, From 2D Toda hierarchy to conformal maps for domains of Riemann sphere, e-print archiv: math.NA/0212361.

9. A.Hurwitz, R.Courant, Vorlesungen /”uber allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie, Springer-Verlag, 1964.

10. I.K.Kostov, I.Krichever, M.Mineev–Weinstein, P.B.Wiegmann, A.Zabrodin, τ-function for analytic curves, Random matrices and their applications, MSRI publication, v.40, Cambridge Acabemic Press, 2001.

11. S.M.Natanzon, Formulas for A_{n} and B_{n}–solutions of WDVV equations, Journal of Geometry and Physics, 39/4, (2001), 323-336.

12. R.Graham, D.Knuth, O.Patashnik, Concrete mathematics, Addison-Wesley publishing Company Reading MA, 1994 (5.63).

13. L.A.Takhtajan, Free boson tau-functions for compact Riemann surfases and closed smooth Jourdan curves. Currents correlation fanctions. Lett. Math. Phys. 56 (2001), 181-228.

14. A.Marshakov, P.B.Wiegmann, A.Zabrodin, Integrable structure of the Dirichlet boundary problem in two dimentions, Commun.Math.Phys.227(2002) 131-153

15. I.Krichever, A.Marshakov, A.Zabrodin, Integrable structure of the Dirichlet boundary problem in multiply-connected domains, e-print archiv: hep-th/0309010.