The Stokes Phenomenon for Some Moment Partial Differential Equations

We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them, we describe Stokes lines, anti-Stokes lines, jumps across Stokes lines, and a maximal family of solutions.


Introduction
In this article, we generalise our results from [18] concerning summability and Stokes phenomenon for the formal solutions of the Cauchy problem for the complex heat equation. In the present paper, we consider the Cauchy problem for general homogeneous linear moment partial differential equation with constant coefficients in two complex variables (t, z) where P (λ, ζ ) is a polynomial of two variables of degree N with respect to λ. Here, ∂ m 1 ,t and ∂ m 2 ,z denote the formal moment differentiations introduced by W. Balser and M. Yoshino [3], which generalise the usual and fractional differentiations.
Such type of equations was previously investigated by the first author [15][16][17] and by A. Lastra, S. Malek, and J. Sanz [11], mainly in the context of multisummability in a given direction. Now, we use the similar methods as in the abovementioned papers to the study of multisummable normalised formal solution u of Eq. 1. It means that u has to be multisummable in every direction but finitely many singular directions. For this reason, we assume that the Cauchy data have finitely many singular or branching points z 0 , . . . , z n ∈ C \ {0} and are analytically continued to C \ n j =0 {z j t : t ≥ 1}, and that satisfy the appropriate exponential growth condition at the infinity. Observe that by the linearity of Eq. 1, it is sufficient to consider the case when there is exactly one such point, say z 0 ∈ C \ {0}. Therefore, we only consider the case when ϕ j (z) ∈ O( C \ {z 0 }).
Using such formal multisummable solution u, for any nonsingular admissible multidirection d, we are able to construct its multisum u d . This multisum is an actual solution of Eq. 1 as a holomorphic function in some sectorial neighbourhood of the origin.
The main purpose of this article is the description of these actual solutions and the study of the relations between them. To this end, we introduce the concept of maximal family of solutions. It is defined as the whole family of actual solutions, which can be obtained by the method of multisummability.
The relations between solutions are studied in the context of the Stokes phenomenon. It means that we find the Stokes lines, which separate different actual solutions constructed from the same multisummable formal power series solution. We also calculate the differences between actual solutions on such lines, which are called jumps across the Stokes lines.
To study such jumps, we apply the Laplace-type hyperfunctions supported on the Stokes line.
In this way, we get the main result of the paper about the maximal family of solutions and the Stokes phenomenon for Eq. 1, which is given in Theorem 3.
In the special case when ∂ m 1 ,t and ∂ m 2 ,z are replaced by ∂ t and ∂ z , we get the description of the Stokes phenomenon for general linear PDEs with constant coefficients.
In this sense, the paper gives the application of theory of summability for PDEs to the description of maximal family of solutions and to the study of Stokes phenomenon for such equations.
The paper is organised as follows. Section 2 consists of basic notations. In Section 3, we recall Balser's theory of moment summability. In particular, we introduce kernel functions and connected with them moment functions, Gevrey order, moment Borel and Laplace transforms, k-summability, and multisummability. In the next section, we recall the concept of moment differential operators and their generalisation to pseudodifferential operators. In Section 5, we recall the notion of Stokes phenomenon. We define Stokes lines and jumps across them for multisummable formal power series. We also introduce Laplace-type hyperfunction on Stokes lines, which allows us to describe these jumps. In Section 6, we introduce the idea of a maximal family of normalised actual solutions of non-Kowalevskian equation. We describe such family of solutions of Eq. 1 in the case when formal solution u is multisummable (Theorem 1). In Section 7, we recall how to reduce the Cauchy problem (1) to a family of the Cauchy problems of simple pseudodifferential equations. Next, using the theory of moment summability, we find the integral representation of actual solutions of these simple pseudodifferential equations in the case when their formal solutions are summable (Proposition 5). It allows us to describe a maximal family of solutions of simple equations, Stokes lines, and jumps across them (Theorem 2). Finally, we return to the equation (1), and using the theory of multisummability, we get the main result of the paper, i.e. the description of a maximal family of solution, Stokes lines, and jumps across them for the equation (1), which is given in Theorem 3. In the last section, we present a few examples of special cases of moment partial differential equations with constant coefficients, where by using hyperfunctions we derive the form of jumps across obtained Stokes lines.

Notation
A sector S in a direction d ∈ R with an opening α > 0 and a radius R ∈ R + in the universal covering spaceC of C \ {0} is defined by the following: This sector is called unbounded if R = +∞ and the notation S = S d (α) will be used. If the opening α is not essential, the sector S d (α) is denoted briefly by S d .
A complex disc D r in C with a radius r > 0 is a set of the form In case that the radius r is not essential, the set D r will be designated briefly by D. We also denote briefly a disc-sector S d (α) ∪ D (resp. S d ∪ D) by S d (α) (resp. S d ).
If a function f is holomorphic on a domain G ⊂ C n , then it will be denoted by f ∈ O(G). Analogously, the space of holomorphic functions of the variable z 1/γ = (z 1/γ 1 1 , . . . , z 1/γ n n ) on a domain G ⊂ C n is denoted by O 1/γ (G), where z = (z 1 , . . . , z n ) ∈ C n , γ = (γ 1 , . . . , γ n ) ∈ N n and 1/γ = (1/γ 1 , . . . , 1/γ n ). In other words, f ∈ O 1/γ (G) if and only if the function w → f (w γ ) is analytic for every w γ = (w γ 1 1 , . . . , w γ n n ) ∈ G. More generally, if E denotes a complex Banach space with a norm · E , then by O(G, E) (resp. O 1/γ (G, E)), we shall denote the set of all E-valued holomorphic functions (resp. holomorphic functions of the variables z 1/γ ) on a domain G ⊆ C n . For more information about functions with values in Banach spaces, we refer the reader to [2,Appendix B]. In the paper, as a Banach space E, we will take the space of complex numbers C (we abbreviate The space of formal power series ∞ n=0 a n t n with a n ∈ E is denoted by E[[t]]. We use the "hat" notation ( u, u i , f ) to denote the formal power series. If the formal power series u (resp. u i , f ) is convergent, we denote its sum by u (resp. u i , f ).

Definition 1
Suppose k ∈ R, S is an unbounded sector and u ∈ O 1/γ (S, E). The function u is of exponential growth of order at most k, if for every proper subsector S * ≺ S (i.e. S * \ {0} ⊆ S) there exist constants C 1 , C 2 > 0 such that u(x) E ≤ C 1 e C 2 |x| k for every x ∈ S * . If this is so, one can write u ∈ O k 1/γ (S, E) and u ∈ O k 1/γ (C, E) for S = C. More generally, if G is an unbounded domain in C n and u ∈ O 1/γ (G, E), then u ∈ O k 1/γ (G, E) if for every set G * satisfying G * ⊂ Int G there exist constants C 1 , C 2 > 0 such that u(x) E ≤ C 1 e C 2 |x| k for every x ∈ G * .

Kernel and Moment Functions, K-Summability, and Multisummability
In this section, we recall the notion of moment methods introduced by Balser [2]. It allows us to describe moment Borel transforms, Gevrey order, Borel summability, and multisummability.
Definition 2 (see [2,Section 5.5]) A pair of functions e m and E m is said to be kernel functions of order k (k > 1/2) if they have the following properties: 1. e m ∈ O(S 0 (π/k)), e m (z)/z is integrable at the origin, e m (x) ∈ R + for x ∈ R + and e m is exponentially flat of order k as z → ∞ in S 0 (π/k) (i.e., for every ε > 0 there exist 4. Additionally, we assume that the corresponding moment function satisfies the normalisation property m(0) = 1. In case k ≤ 1/2 the set S π (2π −π/k) is not defined, so the second property in Definition 2 can not be satisfied. It means that we must define the kernel functions of order k ≤ 1/2 and the corresponding moment functions in another way. To this end, we use the ramification at z = 0.
Definition 3 (see [2, Section 5.6]) A function e m is called a kernel function of order k > 0 if we can find a pair of kernel functions e m and E m of order pk > 1/2 (for some p ∈ N) so that e m (z) = e m z 1/p /p for z ∈ S 0 (π/k).
For a given kernel function e m of order k > 0, we define the corresponding moment function m of order 1/k > 0 by Eq. 2 and the kernel function E m of order k > 0 by Eq. 3.

Remark 2
Observe that by Definitions 2 and 3 we have z j m (jp) .
As in [16], we extend the notion of moment functions to real orders.

Definition 4
We say that m is a moment function of order 1/k < 0 if 1/m is a moment function of order −1/k > 0. We say that m is a moment function of order 0 if there exist moment functions m 1 and m 2 of the same order 1/k > 0 such that m = m 1 /m 2 .
By Definition 4 and by [2, Theorems 31 and 32] we have the following: Proposition 1 Let m 1 , m 2 be moment functions of orders s 1 , s 2 ∈ R respectively. Then we have as follows: • m 1 m 2 is a moment function of order s 1 + s 2 .
• m 1 /m 2 is a moment function of order s 1 − s 2 .
Example 1 For any k > 0, the classical kernel functions and the corresponding moment function, satisfying Definition 2 or 3, are given by the following: They are used in the classical theory of k-summability.
Example 2 For any s ∈ R, we will denote by s the function as follows: Observe that by Example 1 and Definition 4, s is an example of a moment function of order s ∈ R.
The moment functions s will be extensively used in the paper, since every moment function m of order s has the same growth as s . Precisely speaking, we have the following: Proposition 2 (see [2,Section 5.5]) If m is a moment function of order s ∈ R, then there exist constants a, A, c, C > 0 such that ac n s (n) ≤ m(n) ≤ AC n s (n) for every n ∈ N 0 .
Using Balser's theory of general moment summability ([2, Section 6.5], in particular [2, Theorem 38]), we apply the moment functions to define moment Borel transforms, the Gevrey order and the Borel summability. We first introduce the following: is called an m-moment Borel transform.
We define the Gevrey order of formal power series as follows: Definition 7 Let e m , E m be a pair of kernel functions of order 1/k > 0 with a moment function m and let d ∈ R.
, then the integral operator T m,d defined as follows: for some ε, R > 0, then the integral operator T − m,d is defined as follows: is the boundary of a sector contained in S d ( π k + ε, R) with bisecting direction d, a finite radius, an opening slightly larger than π/k, and the orientation is negative) is called an inverse m-moment Laplace transform in a direction d.
Now, we are ready to define the summability of formal power series.
Moreover, the k-sum of u in the direction d is given by the following: (4) is k-summable in all directions d but (after identification modulo 2π ) finitely many directions d 1 , . . . , d n , then u is called k-summable and d 1 , . . . , d n are called singular directions of u.
Next, we extend the notion of k-summable formal power series to that which are multisummable.

Definition 10
Let k 1 > · · · > k n > 0 and let κ 1 , . . . , κ n be defined by Remark 6 Admissibility of d with respect to k is equivalent to the inclusions Definition 11 Let m 1 , . . . , m n be moment functions of positive orders respectively 1/κ 1 , . . . , 1/κ n , where κ 1 , . . . , κ n are constructed in Definition 10. A formal power series is called k-multisummable in the admissible multidirection, provided by the following: Moreover, the k-multisum of u in the multidirection d is given by the following:

Definition 13
If u has at most (after identification modulo 2π ) finitely many singular directions of each level k j , 1 ≤ j ≤ n, then u is called k-multisummable.
Moreover, if additionally u j is k j -summable with n j singular directions d j,1 , . . . , d j,n j (for j = 1, . . . , n) then u is k-multisummable and d j,1 , . . . , d j,n j are singular directions of u of level k j .

Moment Operators
In this section, we recall the notion of moment differential operators constructed by Balser and Yoshino [3] and the concept of moment pseudodifferential operators introduced in the previous papers of the first author [15,16].

Definition 14
Let m be a moment function. Then, the linear operator ∂ m , x : ] is defined by the following: and is called an m-moment differential operator ∂ m,x .
Below, we present most important examples of moment differential operators. Other examples, including also integro-differential operators, can be found in [16,Example 3].
where ∂ s x denotes the Caputo fractional derivative of order s defined by the following: Immediately by the definition, we obtain the following connection between the moment Borel transform and the moment differentiation.

Proposition 3 Let m and m be two moment functions. Then, the operators B m , ∂ m,t : E[[t]] → E[[t]] satisfy the following commutation formulas for every u ∈ E[[t]]
and for m = mm : Now, following [16], we generalise moment differential operators to a kind of pseudodifferential operators. Namely, we have the following: Definition 13]) Let m be a moment function of order 1/k > 0 and λ(ζ ) be an analytic function of the variable ξ = ζ 1/γ for |ζ | ≥ r 0 (for some γ ∈ N and r 0 > 0) of polynomial growth at infinity. A moment pseudodifferential operator λ(∂ m,z ) : is defined by the following: γ |w|=ε means that we integrate γ times along the positively oriented circle of radius ε. Here, the integration in the inner integral is taken over a ray {re iθ : r ≥ r 0 }. [15,Definition 9] Let λ(ζ ) be an analytic function of the variable ξ = ζ 1/γ for |ζ | ≥ r 0 (for some γ ∈ N and r 0 > 0) of polynomial growth at infinity. Then, we define the pole order q ∈ Q and the leading term λ 0 ∈ C \ {0} of λ(ζ ) as the numbers satisfying the formula lim ζ →∞ λ(ζ )/ζ q = λ 0 . We write it also λ(ζ ) ∼ λ 0 ζ q .

Stokes Phenomenon and Hyperfunctions
Now, we extend the concept of the Stokes phenomenon (see [18,Definition 7]) to multisummable formal power series u ∈ E[[t]].

Definition 17 Assume that u ∈ E[[t]]
is k-multisummable with singular directions d j,1 , . . . , d j,n j of level k j , 1 ≤ j ≤ n. Then, for every l = 1, . . . , n j and j = 1, . . . , n, the set L d j,l = {t ∈C : arg t = d j,l } is called a Stokes line of level k j for u.
Assume now that for fixed j ∈ {1, . . . , n} the vector d = (d 1 , . . . , d n ) is an admissible multidirection with a singular direction d j of level k j and with nonsingular directions d l of level k l for l = j , and let d ± denotes a direction close to d j and greater (resp. less) than d j , and let u We will describe jumps across the Stokes lines in terms of hyperfunctions. The similar approach to the Stokes phenomenon one can find in [8,13,20]. For more information about the theory of hyperfunctions, we refer the reader to [9].
We will consider the space as follows: of Laplace-type hyperfunctions supported by L d with exponential growth of order k. It means that every hyperfunction G ∈ H k (L d ) may be written as follows: Let γ d be a path consisting of the half-lines from e id − ∞ to 0 and from 0 to e id + ∞, i.e.
By the Köthe-type theorem [10], one can treat the hyperfunction G(s) = [g(s)] d as the analytic functional defined by the following: To describe the jumps across the Stokes lines in terms of hyperfunctions, first assume that f ∈ C[[t]] is k-summable, m is a moment function of order 1/k and d is a singular direction. By Eq. 4, the jump for f across the Stokes line L d is given by the following: Observe that we can treat g 0 (t) So, combining Eq. 5 with Eq. 6, we conclude that for sufficiently small r > 0 and t ∈ S d ( π k , r). Now, let f ∈ C[[t]] be k-multisummable and d be as in Definition 17 with L d j being the Stokes line of level k j . We additionally assume as in Remark 7 that f = f 1 + · · · + f n , where f j is k j -summable. Then, by Remark 7, analogously as in the summable case, the jump across L d j of level k j is given by the following: and we may describe this jump in terms of hyperfunctions as in the previous case. Similarly , then we are able to describe jumps for u(t, z) at the point z = 0 in terms of hyperfunctions. Namely, we have the following: Analogously, we calculate jumps across a Stokes line L d j of level k j for kmultisummable u satisfying u = u 1 + · · · + u n , where u i is k i -summable (i = 1, . . . , n).

Remark 9
In some special cases, we are also able to describe jumps for u(t, z) at any point z ∈ D. It is possible in the case when u is a multisummable solution of instead of Eq. 1. In this case, we are able to reduce the problem of description of jumps for u(t, z) at the fixed point z ∈ D, to the problem of description of jumps for the auxiliary formal power series u z (t, s) := u(t, s + z) at the point s = 0. Since the derivative operator ∂ z is invariant under the translation, i.e. (∂ z u)(t, s + z) = ∂ s ( u(t, s + z)), we conclude that u z (t, s) is a multisummable solution of P (∂ m 1 ,t , ∂ s )u z = 0 ∂ j m 1 ,t u z (0, s) = ϕ z,j (s) := ϕ j (s + z) for j = 0, . . . , N − 1.
Since in general the moment differential operators are not invariant under translation, we are not able to use this method to describe the jumps for solutions of P (∂ m 1 ,t , ∂ m 2 ,z )u = 0 at any point z ∈ D.

A Maximal Family of Solutions
Now, we are ready to describe a family of normalised actual solutions of given non-Kowalevskian equation using sums of multisummable formal power series solution. More precisely, we consider the Cauchy problem where m 1 , m 2 are moment functions of orders s 1 , s 2 > 0 respectively and is a general polynomial of two variables, which is of order N with respect to λ.
If P 0 (ζ ) defined by Eq. 9 is not a constant, then a formal solution of Eq. 8 is not uniquely determined. To avoid this inconvenience, we choose some special solution which is already uniquely determined. To this end, we factorise the polynomial P (λ, ζ ) as follows: where λ 1 (ζ ), . . . , λ l (ζ ) are the roots of the characteristic equation P (λ, ζ ) = 0 with multiplicity N 1 , . . . , N l (N 1 + · · · + N l = N ) respectively.
Hence, λ α (∂ m 2 ,z ) are well-defined moment pseudodifferential operators and consequently also the operator Under the above assumption, by a normalised formal solution u of Eq. 8, we mean such solution of Eq. 8, which is also a solution of the pseudodifferential equation P (∂ m 1 ,t , ∂ m 2 ,z ) u = 0 (see [15,Definition 10]).
Since the principal part of the pseudodifferential operator P (∂ m 1 ,t , ∂ m 2 ,z ) with respect to ∂ m 1 ,t is given by ∂ N m 1 ,t , the Cauchy problem (8) has a unique normalised formal power series solution u ∈ O(D) [[t]]. If we additionally assume that u is multisummable, then using the procedure of multisummability in nonsingular directions, we obtain a family of normalised actual solutions of Eq. 8 on some sectors with respect to t. This motivates us to introduce the following definitions.

Definition 18 Let S be a sector in the universal covering spaceC. A function u ∈ O(S ×D) is called a normalised actual solution of Eq. 8 if it satisfies
In [18], we introduced a maximal family of solutions of Eq. 8 in the case when a formal power series solution is k-summable. It is a collection of all actual solutions of Eq. 8 constructed by the procedure of k-summability. Now, we generalise this definition to the multisummable case.

Definition 19
Assume that the normalised formal power series solution u of Eq. 8 is kmultisummable, J is a finite set of indices, and V is a sector with an opening greater than π/k n on the Riemann surface of t 1 q for some q ∈ Q + . We say that {u i } i∈J with u i ∈ O(V i × D) is a maximal family of solutions of Eq. 8 on V × D if the following conditions hold: (a) V i ⊆ V is a sector of opening greater than π/k 1 for every i ∈ J . Theorem 1 Let u be a k-multisummable normalised formal power series solution of Eq. 8 with a k-multisum in a nonsingular admissible multidirection d given by u d = S k,d u and satisfying u = u 1 + · · · + u n , where u j is k j -summable for j = 1, . . . , n. Assume that there exists q ∈ Q + , which is the smallest positive rational number such that u d (t, z) = u d (te 2qπi , z) for every nonsingular multidirection d. Suppose that the set of singular directions of u of level k j modulo 2qπ is given by {d j,1 , . . . , d j,n j }, where 0 ≤ d j,1 < · · · < d j,n j < 2qπ (j = 1, . . . , n).
It means that for every l ∈ J , the function u l is well-defined.
To show (ii), observe that for every sufficiently small ε > 0 there exists r > 0 such that S k j ,d j u j is analytically continued to the set as follows: Hence, the whole function u l is analytically continued to the set V l (ε, r) × D. Finally, we prove (iii). Since the inequality |I 1,l 1 ∩· · ·∩I n,l n | > π k 1 holds for every l ∈ J , we are able to take such small ε > 0 that the opening of V l (ε, r) (V l for short) is greater than π k 1 . We claim that {V l } l∈J is a covering of W r . To this end, we take any t ∈ W r . Then, we may choose l ∈ N n such that 1 ≤ l j ≤ n j and arg t ∈ [d j,l j , d j,l j +1 ) for j = 1, . . . , n. For such choice of l there exists δ > 0 such that arg t − π 2k j + δ 2 , arg t + π 2k j + δ ⊂ I j,l j ∩ · · · ∩ I n,l n for j = 1, . . . , n.
It means that |I j,l j ∩ · · · ∩ I n,l n | ≥ π k j + δ 2 > π k j for j = 1, . . . , n, so l ∈ J and t ∈ V l . By the freedom of choice of t ∈ W r , {V l } l∈J is a covering of W r .
By the moment version of [1, Theorem 6.2], we conclude that the space of kmultisummable series in a multidirection d is a moment differential algebra over C. It means that it is a linear space, which is also closed under multiplication and moment differentiations, and which for any k-multisummable series f and g satisfies: Hence, Additionally, since u(t, z) = ∞ j =0 u j (z)t j on V l × D, by [2,Proposition 8] and by the definition of multisummable series we get the following: Therefore u l is an actual solution of Eq. 8 for l ∈ J . Now, assume that V l ∩ V˜l = ∅ and u l ≡ u˜l on (V l ∩ V˜l) × D for some l,l ∈ J and l =l. It means that there exists admissible multidirections d = (d 1 , . . . , d n ), d j ∈ (d j,l j , d j,l j +1 ) andd = (d 1 , . . . ,d n ),d j ∈ (d j,l j , d j,l j +1 ) such that u l = S k 1 ,d 1 u 1 + · · · + S k n ,d n u n = S k 1 ,d 1 u 1 + · · · + S k n ,d n u n = u˜l.
Since S k j ,d j u j and S k j ,d j u j are both analytic on the non-empty by the Relative Watson's lemma [14, Proposition 2.1], we conclude that S k j ,d j u j = S k j ,d j u j for j = 1, . . . , n. Since l =l, without loss of generality, we may assume that This contradicts the fact that d i,l i is a singular direction of level k i . So if V l ∩V˜l = ∅ then u l ≡ u˜l on (V l ∩V˜l) × D for every l,l ∈ J , l =l. By the construction of the family {u l } l∈J , the last two conditions in Definition 19 are also satisfied, which completes the proof.

General Linear Moment Partial Differential Equations with Constant Coefficients
We will study the Stokes phenomenon and the maximal family of solutions for the normalised formal solution u of Eq. 8. Let us recall that we may reduce the Cauchy problem (8) of a general linear moment partial differential equation with constant coefficients to a family of the Cauchy problems of simple moment pseudodifferential equations. Namely, we have Moreover, if q α is a pole order of λ α (ζ ) and q α = max{0, q α }, then For this reason, we will study the following simple moment pseudodifferential equation where m 1 , m 2 are moment functions of orders respectively s 1 , s 2 > 0 such that qs 2 > s 1 , γ ∈ N, λ(ζ ) ∼ λ 0 ζ q with q = μ/ν for some relatively prime μ, ν ∈ N satisfying qγ ∈ N. We start from the following representation of summable solutions of Eq. 11.
Moreover, the sets L δ l and L δ l ± π 2K (l = 0, . . . , μ − 1) are respectively Stokes lines and anti-Stokes lines for u. The jump across the Stokes line L δ l is given by the following: and v(s, z) = B m u(s, z) has the representation (14).
Proof Since P (λ, ζ ) is given by Eq. 15, by Proposition 4, a normalised formal solution u of Eq. 8 may be written as u = u 0 + u 1 + · · · + u n , where u 0 is a convergent power series solution of the pseudodifferential equation (16) and u i is a 1/K i -Gevrey power series solution of Eq. 17 with the initial data having the same holomorphic properties as ϕ j (z). More precisely, by Proposition 4 we conclude that u i = l i α=1 N iα β=1 u iαβ , where u iαβ is a formal solution of a simple pseudodifferential equation as follows: where ϕ iαβ (z) := and d iαβj (ζ ) are some holomorphic functions of the variable ξ = ζ 1/γ and of polynomial growth. Since ϕ j (z) ∈ O q n K n ( C \ {z 0 }) and q i K i ≤ q n K n , we see that Hence, by Proposition 5 and Theorem 2, we see that u iαβ is K i -summable with the singular directions given by q i arg z 0 + 2jπ ν i − arg λ iα mod 2πq i for j = 0, . . . μ i − 1. Consequently, u is K-multisummable in any nonsingular admissible multidirection d =  (d 1 , . . . , d n ). Since a formal power series u 0 is convergent, its sum u 0 is well-defined and by Remark 7, we conclude that K-multisum S K,d u of u is given by Eq. 18, so (a) holds.
Since the set of singular directions of order K i is given by i , we get the description of Stokes lines L δ i,j and anti-Stokes lines L δ i,j ± π 2K i of level K i for δ i,j ∈ i and i = 1, . . . , n, so (e) is also satisfied.
Finally, to obtain (f) by Theorem 2, we calculate the jumps for u across the Stokes lines L d i of level K i . Using Remark 9, we get (f').
Let us illustrate our theory on the following simple example.