## Abstract

In this paper, we present a method solving the problem of the asymptotic expansion of the integral , in the case when *D* is a bounded domain in (*n*≥2), and the set *S* of stationary points of the phase *f* is a hypersurface. This problem was considered in the literature, in the two-dimensional case, where it is required that the Laplacian △*f* of the phase *f* does not vanish on *S*, and the curve *S* cuts transversely ∂*D*. It will be seen that the order of degeneracy of normal derivatives of *f*, with respect to the surface *S*, plays a key role in solving the problem. We shall develop complete asymptotic expansions when this order is constant along *S*, and show that the problem leads to the use of special functions in the other case.

## 1. Introduction

Consider the asymptotic expansion of the integral , where *D* is a bounded domain in (*n*≥2), and *f* and *g* are smooth real-valued functions on the closure of *D*. It is well known that the major contributions to the asymptotic expansion of *I*(*λ*) come from the stationary points of *f*, from ‘corners’ of the boundary ∂*D* of *D*, and from the points where level sets of *f* are tangent to ∂*D* [1], ch. VIII.

In this paper, we are especially concerned with the contribution of the set of stationary points of *f*, in the case when *S* is a hypersurface of This problem was considered in the literature, in the two-dimensional case, *S* is then a simple curve [2,3], because of its theoretical and practical importance (see [4,5] and the introduction in Kontorovitch *et al*. [2]). In these works, one is restricted to the case where the curve *S* cuts transversely the boundary ∂*D* of *D* only, and it is required that △*f*(*x*)≠0 for all *x*∈*S*. The case of tangency was solved later in Benaissa & Roger [6]. In this context, it is useful to cite the work of Kaminski [7], which treated the problem in the three-dimensional case when *S* is a curve.

In this paper, we obtain complete asymptotic expansions under the following more general condition.
1.1The choice of condition (1.1) is motivated by the fact that it ensures that a certain function of one real variable *t*, parametrized by the generic element *s* of the hypersurface *S*, has a constant order of degeneracy of its derivatives at *t*=0. On the other hand, the commonly used condition (△*f*)(*x*)≠0 on *S* is a particular case of (1.1), corresponding to *r*=2 (see lemma 2.1).

Our problem can be reduced, by using a partition of unity, to the asymptotic expansion of the integral
1.2where *δ* is a small positive number, and *D*_{δ} is the set of points of whose distance from *S* is not greater than *δ*.

For the sake of simplicity, we restrict ourselves to the case *n*=3, in such a way that *S* becomes a surface in , the techniques developed can be carried out safely to higher dimensions.

One of the advantages of our approach is that the coefficients of the expansions are expressed in the form of integrals of density functions on the surface *S* and on its boundary ∂*S*. Consequently, we can see separately the contribution of each point of *S*, and the influence of the geometry of the surface *S*.

The paper is organized as follows. In §2, we construct adequate coordinates of *D*_{δ}, and prove lemma 2.1, that enables us to analyse the problem relative to the normal derivatives of the phase *f*. In §3, using a tubular neighbourhood of the surface *S*, obtained in the previous section, we construct a complete asymptotic expansion of the integral *I*_{δ}(*λ*), in the case when the surface *S* is without boundary (theorem 3.2). In §4, we consider the case where the surface *S* cuts the boundary ∂*D* of *D*: complete asymptotic expansions are obtained in the case of transversality (theorem 4.1) and in the case of tangency (theorem 4.2). In §5, we show that the coefficients of the asymptotic expansions depend on the mean and the Gaussian curvature of *S* (theorem 5.1). Finally, in §6, we propose a method analysing the problem when condition (1.1) is not satisfied.

## 2. Preliminary transformations

Throughout this paper, we use the notation
for every surface *W* in , *y*_{x} being the unit normal to *W* at *x*. Here, *T*_{δ}(*W*) is a tubular neighbourhood of *W* and *L*_{δ}(*W*) is its edge (see Cannas da Silva [8]). Note that, for *δ* small enough, it is useful to think of |*t*| as the distance from the point *x*+*ty*_{x} to the surface *W* [3], p. 52.

Because *f* and *g* are smooth functions in , and *S* is a smooth surface, then by a classical theorem, we can assume that *f* and *g* are extended to smooth functions in some open bounded neighbourhood *V* of , and that the surface *S* is extended to a smooth surface, denoted by , so that *S* is included in the interior of the surface [9], theorem 4.1 p. 10. Obviously, we may choose so that for *δ* sufficiently small , being the closure of These extensions are not needed in the situation where the surface *S* is without boundary. For every *x*∈*S*, there are two open bounded subsets *O*_{x}, of such that being the closure of *O*_{x}), and a local parametrization of
such that *φ*_{x}(0,0)=*x*. By the use of a Rodrigues' formula, we may choose *φ*_{x} such that ∂*φ*_{x}/∂*u*_{1} and ∂*φ*_{x}/∂*u*_{2} are principal directions of the surface *S* [10], ch. 4, §16. Thus, because of the compactness of *S*, there are a finite number of elements *x*_{1},…,*x*_{N} of *S* such that *S*⊂*φ*_{x1}(*O*_{x1})∪⋯∪*φ*_{xN}(*O*_{xN}). Let us define
2.1
2.2
and
2.3Obviously, , and is a partition of *S*. On the other hand, a tubular neighbourhood theorem ([8], homework 5, p. 41) implies that for *δ*>0 sufficiently small, the map: *S*×[−*δ*,*δ*]∋(*x*,*t*) , is one-to-one of *S*×[−*δ*,*δ*] onto *T*_{δ}(*S*). We deduce that (1≤*l*≤*N*) is a partition of *T*_{δ}(*S*). Hence, we have for *δ* sufficiently small
2.4Note that *δ* is also so that *T*_{δ}(*S*) is included in the neighbourhood *V* of , defined earlier (because the extended functions *f* and *g* must be defined on *T*_{δ}(*S*)). In the case where the surface *S* is without boundary, we must have *T*_{δ}(*S*)⊂*D*.

Consider, for each 1≤*l*≤*N*, the transformation
2.5where , and *y*(*u*) is the unit normal to *S* at the point . The Jacobian matrix of the transformation *M*_{l} is , where , and *y*_{1}=∂*y*(*u*)/∂*u*_{1}, *y*_{2}=∂*y*(*u*)/∂*u*_{2}. By Rodrigues' formula, we have and , where *k*_{1} and *k*_{2} are the principal curvatures of the extended surface . Therefore, noting that , and *y* are orthogonal, the Jacobian ∂*x*/∂(*u*,*t*) of the transformation *M*_{l} is given by
where *H* and *K* are the mean and the Gaussian curvature of , respectively. It follows that for all , and using the compactness of one can choose *δ* so that ∂*x*/∂(*u*,*t*)(*u*,*t*)>0 for all and |*t*|≤*δ*. On the other hand, the tubular neighbourhood theorem implies that, for *δ* sufficiently small, *M*_{l} is a diffeomorphism of the set {(*u*,*t*):*u*∈*O*_{xl},|*t*|≤*δ*} onto its image. Thus, the transformation yields
where , *Θ*_{l}(*u*,*t*)=[1−2*tH*+*t*^{2}*K*]*G*_{l}(*u*,*t*) and *ds* is the element surface of *S* at the point ,
2.6and
2.7Using the inverse of the Jacobian matrix of the transformation *M*_{l} and the relation [10, ch. 4, §4], we obtain
2.8Lemma 2.1 transforms condition (1.1) to a condition related to the order of degeneracy of the derivatives at *t*=0, of the functions of one real variable

### Lemma 2.1

(1)

*Condition (1.1) is equivalent to:*2.9(2)

*If condition (1.1) holds, then we have*2.10*for all 1*≤*l*≤*N*,*and φ*_{xl}(*u*)∈*S*.(3)

*If the integer r is even, then the quantities*(∂^{r}*F*_{l}/∂*t*^{r})(*u*,0)*are of the same sign*(≤0) or (≥0).

### Proof.

We prove that condition (1.1) implies condition (2.9). For the converse, we may use similar arguments. By the chain rule formula, we have
Then, by taking into account that *f* is constant on *S*, we obtain for all *φ*_{xl}(*u*)∈*S*
2.11Using (2.8) and noting that the components (*y*^{1},*y*^{2},*y*^{3}) of the unit normal *y* do not vanish simultaneously, we deduce from (2.11) and the fact that (∂*f*/∂*x*_{j})(*x*(*u*,0))=0 (1≤*j*≤3) that (∂*F*_{l}/∂*t*)(*u*,0)=0. Differentiating again with respect to *x*_{j}, we see that is a sum of terms, one of which is ∂^{2}*F*_{l}/∂*t*^{2}(∂*t*/∂*x*_{j})^{2}, and each of other terms is a product of terms, one of which is in the form with *ϱ*=0 or *ϱ*=1. If *ϱ*=0, the fact that *f* is constant over *S* implies that . If *ϱ*=1, we deduce from (∂*F*_{l}/∂*t*)(*u*,0)=0 that . Repeating this argument *r* times, we obtain
2.12and
2.13for all *φ*_{xl}(*u*)∈*S*. Formula (2.9) follows from (1.1), (2.12) and (2.13). Formula (2.10) results from formulae (2.8) and (2.13), and the fact that (*y*^{1})^{2}+(*y*^{2})^{2}+(*y*^{3})^{2}=∥*y*∥^{2}=1. Assertion three results from (2.13). □

### Remark 2.2

Lemma 2.1 shows that condition (1.1) is invariant under change of coordinates.

Formula (2.10) shows that the function (∂^{r}*F*_{l}/∂*t*^{r})(*u*,0) does not vanish for *φ*_{xl}(*u*)∈*S*. In the rest of this paper, we assume, without mentioning it each time, that the following conditions are verified.

(

*C*_{1}) (∂^{r}*F*_{l}/∂*t*^{r})(*u*,0)>0 for all 1≤*l*≤*N*, and*φ*_{xl}(*u*)∈*S*.(

*C*_{2}) The integer*r*is even. (Obviously, techniques used stay valid for*r*odd).

Now, we establish a useful property for the functions *F*_{l} (1≤*l*≤*N*). Formula (2.9) allows us, using a Taylor expansion, to write for all *φ*_{xl}(*u*)∈*S* and *t* small enough,
where
is of class and
2.14Thus, taking into account (2.7) and the fact that *S* is included in the union of the sets *φ*_{xl}(*O*_{xl}) (1≤*l*≤*N*), one deduces that for *δ* small enough *f*(*x*) does not vanish over the edge *L*_{δ}(*S*) of the tubular neighbourhood *T*_{δ}(*S*) of *S*. Because for *δ* small enough, *L*_{δ}(*S*) and *S* are disjoint compact subsets of *V* , then there is a closed neighbourhood *V* _{1}⊂*V* of *L*_{δ}(*S*) having no intersection with *S*, such that
2.15Because we are interested in the contribution to the integral *I*(*λ*) from the stationary surface *S*, we may assume that in (1.2) the function *g*(*x*) vanishes outside a neighbourhood *V* _{2} of *S*: for technical convenience we assume that
2.16for all , where *V* _{2}={*x*∈*V* :|*f*(*x*)|^{1/r}<*C*/3}.

## 3. The case where the surface *S* is without boundary

We begin this section by proving a lemma providing the asymptotic expansions of the integrals . To do this, we first integrate with respect to the variable *t*, where by the use of a stationary phase method [1, pp. 76–81], we obtain uniform asymptotic expansions with respect to the variable *u*, the desired expansions are then obtained by integrating with respect to the variable *u*.

### Lemma 3.1

*Assume that the set S of stationary points of the function f is a surface with or without boundary, and that condition (1.1) is fulfilled. Then, for δ small enough and g satisfying (2.16), the integrals* (1≤*l*≤*N*), *defined in* (2.4), *have the asymptotic expansions*
3.1*where* *for n odd, and*
*for n even, with* , *ϖ*_{(u)} *being defined, for u*∈ *and* 0≤*ς*≤*ρ*(*u*)=*F*(*u*,*δ*)^{1/r}, *by the relation* *F*(*u*,*ϖ*_{(u)}(*ς*))^{1/r}=*ς*.

### Proof.

In the following proof, *F* denotes the function *F*_{l} defined by (2.7). Consider the transformation
3.2where (see the beginning of §2). The Jacobian of the transformation *Λ* is given by
It is readily seen from (2.1) to (2.3), that . Thus, (2.14) gives
for all *u* in the compact set . As a consequence, we may choose *δ* small enough such that
for all , 0≤*t*≤*δ*. Here, (*F*_{(u)})^{1/r} is the function of one variable
It follows that, for every the function of one variable (*F*_{(u)})^{1/r} is a bijection from [0,*δ*] onto [0,(*F*(*u*,*δ*))^{1/r}], and therefore its inverse function
is defined on the interval [0,*ρ*(*u*)], *ρ*(*u*)=(*F*(*u*,*δ*))^{1/r}. Then, in particular, the transformation *Λ* is one-to-one on the set . Thus, the change of variable , defined by (3.2), and Fubini's formula yield
where
On the other hand, condition (2.16) implies that for *ς*≥*C*/3, where *C*/3<*C*≤*ρ*(*u*). Hence, we are allowed to use formulae (3.13) and (3.14) in Wong [1], pp. 76–81 to obtain
where
Because the derivatives (with respect to *ς*) of *G*_{l}(*u*,*t*) and *ϖ*_{(u)} are bounded, we have
uniformly with respect to *u*. Finally, integrating with respect to *u*, we obtain
where
To approximate the integrals
we set *ι*=−*t* and use similar techniques. □

Now, we are ready to present the main result in this section.

### Theorem 3.2

(a)

*If condition (1.1) is fulfilled, the surface S is with or without boundary, then we obtain for δ sufficiently small and g satisfying condition (2.16), the asymptotic expansion*3.3*where b*_{n}*is zero for n odd, whereas for n even, b*_{n}*is given by*3.4d*s being the surface element of the surface S,**and the coefficients (b*_{n})*may be expressed in terms of the derivatives of the original functions f and g over the surface S. In particular*3.5(b)

*If, in addition, the surface S is without boundary, then the integral I*_{δ}*(λ) has the same expansion (given by (3.3)–(3.5)).*

*Note that, for n=r=2, formula (3.5) corresponds to formula (2.5) in McClure & Wong* [3].

### Proof.

Assertion (a) Results from (2.4) and lemma 3.1.

(b) Because

*S*is a compact subset of*D*, we may choose*δ*sufficiently small such that*T*_{δ}(*S*) is included in*D*. It follows that*D*_{δ}=*T*_{δ}(*S*).

□

## 4. The case when *S* is with boundary

In this section, we assume that the boundary ∂*S* of the surface *S* is a simple smooth closed curve that coincides with the intersection of *S* and ∂*D*, and the boundary ∂*D* of the domain *D* is smooth near ∂*S*. If ∂*D* has corner points in ∂*S*, it suffices to integrate separately on both sides of the surface *S*. We also assume that the extended surface is parametrized in a neighbourhood of ∂*S* by a map
where *L* is the total length of ∂*S*, the two segments {0}×[−*η*,*η*] and {*L*}×[−*η*,*η*] are identified, and the following condition is verified.
For such construction, see [10], ch. 2, §16 and ch. 4, §23.

Let us define, for *δ*>0, the two sets (figure 1)
We have
It follows
Consequently
4.1The asymptotic expansion of the first integral in the right-hand side of (4.1) is given by theorem 3.2. On the other hand, it will be seen that the expansion of the bracketed quantity in the right-hand side of (4.1) may be obtained by similar method pursued in McClure & Wong [3].

As in the definition of the transformations *M*_{l} (see formula (2.5)), define for 0≤*v*_{1}≤*L*, |*v*_{2}|≤*η*,|*t*|≤*δ*, the transformation
*y* being as previously the unit normal to the surface *S* at the point . Again from the tubular neighbourhood theorem, we see that for *δ* small enough, is a diffeomorphism of the rectangular cuboid
onto its image, denoted by *D*_{η,δ} (the two rectangles {0}×[−*η*,*η*]×[−*δ*,*δ*] and {*L*}×[−*η*,*η*]×[−*δ*,*δ*] being identified). Furthermore, because the vectors , and *y* are linearly independent, the Jacobian ∂*x*/∂(*v*_{1},*v*_{2},*t*) of the transformation does not vanish over *R*_{η,δ} for *δ* small enough.

### (a) The case of transversality of ∂*D* and *S*

We assume in this subsection that the boundary ∂*D* of the domain *D* and the surface *S* are not tangent at any point of their intersection ∂*S*. That is what was considered in earlier studies [2,3] in the two-dimensional case.

is defined in a neighbourhood of every point by an equation
4.2where *Φ* is a smooth function defined in a neighbourhood of the point , whose gradient does not vanish. Furthermore, the non-tangency of ∂*D* and *S* ensures that Then, using an implicit function theorem [11], p. 41, we deduce that is expressed in a neighbourhood of the point by , where is a smooth function in a neighbourhood of the point . Hence, using a compactness argument, we may express for *η*, *δ* sufficiently small, (∂*D*∩*D*_{η,δ}) by the equation
where *a*(*v*_{1},*t*) is a smooth function, *a*(*v*_{1},0)=0, and shrinking *δ* if necessary,
for all (*v*_{1},*t*)∈[0,*L*]×[−*δ*,*δ*] (figure 2). It follows that and are included in . Changing variables from *x*=(*x*_{1},*x*_{2},*x*_{3}) to (*v*,*t*)=(*v*_{1},*v*_{2},*t*), the bracketed quantity in (4.1) takes the form
4.3where and Moreover, and can be expressed as
4.4and
4.5respectively. We deduce
and being the first and the second integral in (4.3), respectively. It follows
4.6Now, we transform the integral in (4.6) to a one-dimensional Fourier integral, using the idea in Jones & Kline [12]. Let , with sgn *z*=sgn *t* and consider the change of variables
4.7By means of similar arguments used for the functions *F*_{l} (1≤*l*≤*N*) (§2) and for the transformation *Λ* (§3), we see that *δ* and *η* can be chosen small enough, so that
for every (*v*_{1},*v*_{2},*t*)∈[0,*L*]×[0,*η*]×[−*δ*,*δ*], where is a smooth function >0. Consequently, for *η* and *δ* small enough, in (4.7), *z* is of the form , the transformation is one-to-one on the set [0,*L*]×[−*η*,*η*]×[−*δ*,*δ*] (the two rectangles {0}×[−*η*,*η*]×[−*δ*,*δ*] and {*L*}×[−*η*,*η*]×[−*δ*,*δ*] being identified) and its Jacobian does not vanish.

Let *ρ* be a real positive number such that *C*/3<*ρ*<*C*, the constant *C* being defined in (2.15). For every 0≤*v*_{1}≤*L*, −*ρ*≤*z*≤*ρ*, the set
is a planar curve with extremities *A*_{v1,z}=(*v*_{1},*α*(*v*_{1},*z*),*t*_{1}) and *B*_{v1,z}=(*v*_{1},0,*t*_{2}), where *α*(*v*_{1},*z*) is given by the equation *α*(*v*_{1},*z*)=*a*(*v*_{1},*t*_{1}), the point (*v*_{1},*a*(*v*_{1},*t*_{1}),*t*_{1}) being the intersection of the two planar curves
and

Note that the curve is a part of the curve *γ*_{v1,z}. Furthermore, is included in for *α*(*v*_{1},*z*)≤0 and it is included in for *α*(*v*_{1},*z*)>0 (figure 2). For similar discussion, see [3], p. 53, fig. 3. On the other hand, condition (2.16) implies that
for all |*z*|≥*C*/3 (<*ρ*), with
Changing variables from (*v*,*t*) to (*v*,*z*), the integral in (4.6) takes the form
where
with *ψ*(*z*)=0 for all |*z*|≥(*C*/3)(<*ρ*). Using a Taylor series of the function (of two real variables) and a Maclaurin's expansion of the function (of one real variable) , one may readily get an asymptotic expansion for *Ψ*(*z*) near *z*=0, of the form
where we may verify that, if
4.8for all *n*≤*l*, 0≤*v*_{1}≤*L* (*l* being some integer >0), then
Integrating *Q* times by parts, we obtain
where *χ*(*n*)=0 if *n* is odd and 1 if *n* is even [13], theorem 4.

Now, we express *α*(*v*_{1},*z*) in terms of the function *a*(*v*_{1},*t*) defining the boundary ∂*D* of *D*, and the function Let us define for 0≤*v*_{1}≤*L*, the function *W*_{(v1)}(*t*) by
We have for every 0≤*v*_{1}≤*L*. Hence, the continuity of the function and the compactness of the line segment [0,*L*]×{0} enable us to choose *δ* sufficiently small such that, for every 0≤*v*_{1}≤*L*, the function is one-to-one in the interval [−*δ*,*δ*], and therefore has an inverse smooth function (*W*_{(v1)})^{−1} defined in the interval [*W*_{(v1)}(−*δ*),*W*_{(v1)}(*δ*)]. We may readily verify that the function *α*(*v*_{1},*z*) is defined by
4.9Furthermore, an implicit function theorem implies that *t*=(*W*_{(v1)})^{−1}(*z*) is a smooth function with respect to its arguments (*v*_{1},*z*).

Note that the relation (4.9) allows us to replace the condition (4.8) (related to the function *α*(*v*_{1},*z*)) by the following one (related to the function *a*(*v*_{1},*t*) defining the boundary ∂*D*).
4.10

These calculations and theorem 3.2 allow us to state the following theorem.

### Theorem 4.1

*If S cuts transversely the boundary ∂D of D, S∩∂D=∂S, and condition (1.1) is verified, then we obtain for δ sufficiently small, and g satisfying (2.16), the asymptotic expansion*
*where the following properties are verified*.

(a)

*c*_{0}=*b*_{0}*and c*_{n}=*b*_{n}(*n even*),*b*_{0}*and**b*_{n}*being given by formulae*(3.5)*and*(3.4),*respectively*.(b)

*c*_{n}=*d*_{n}=0,*for odd integers n*.(c)

*c*_{n}*and d*_{n}*represent the contribution of the points in the interior and the points in the boundary ∂S of the surface S, respectively*.(d)

*If the order of contact, at every point s∈∂S, between ∂D and the normal plane to S at the point s, carrying the tangent vector to ∂S at s, is at least equal to l>0 (i.e. formula (4.10) holds), then the coefficients d*._{n}vanish for every n≤l

### (b) Tangency case between *S* and ∂*D*

Here, we consider the same situation as in the previous subsection, except that, instead of transversality of *S* and ∂*D*, we assume that *S* and ∂*D* are tangent at each point of their intersection ∂*S*.

As in §4*a*, is expressed in the vicinity of each point of by equation (4.2), where *Φ* is a smooth function defined in a neighbourhood of the point , whose gradient does not vanish. Because *S* and ∂*D* are tangent at every point (0≤*v*_{1}≤*L*), then
4.11Hence, because the gradient of *Φ* is not zero at the point , we have
4.12Then, an implicit function theorem implies that is defined in a neighbourhood of the point by an equation , where is a smooth function, and Furthermore, (4.2), (4.11) and (4.12) imply that
for all *v*_{1} in a neighbourhood of . Hence, by the same argument as in the transversality case, we may express ∩*D*_{η,δ}) in a neighbourhood of by a smooth function , (*v*_{1},*v*_{2})∈[0,*L*]×[−*η*,*η*], (*η*>0), such that *β*(*v*_{1},0)=0 and (∂*β*/∂*v*_{2})(*v*_{1},0)=0. We assume in the rest of this section that the order of contact between ∂*D* and *S* is constant along ∂*S*. This means that
for all 0≤*n*≤*p*−1, 0≤*v*_{1}≤*L*. The integer *p* is necessarily odd (because *v*_{2} must change sign with *t*).

Using a Maclaurin expansion for the function of one variable , we obtain
where is a smooth function with respect to *v*_{1}, *v*_{2}, and
for every 0≤*v*_{1}≤*L*. Let *T* be the function of one real variable We have
for every 0≤*v*_{1}≤*L*. Hence, by compactness and continuity argument, we deduce that *η*>0 may be chosen sufficiently small, such that for every *v*_{1}∈[0,*L*], the function is one-to-one on the interval [−*η*,*η*] and then has an inverse smooth function ((*T*°*β*_{(v1)}))^{−1} defined on the interval (*T*°*β*_{(v1)})([−*η*,*η*]). So, for each 0≤*v*_{1}≤*L*, *β*_{(v1)} has an inverse function
Consequently, is expressed in a neighbourhood of by the function
where the use of an implicit function theorem implies that the function is of class . Consider, for each 0≤*v*_{1}≤*L*, the function
We have
for all 0≤*v*_{1}≤*L*. It follows that, for every 0≤*v*_{1}≤*L*, *V* _{(v1)} has an inverse smooth function (*V* _{(v1)})^{−1} in a neighbourhood of *t*=0. On the other hand, it is easily seen that
which allows us to write *α*(*v*_{1},*z*) in (4.9), in the form
where is the smooth function defined by

As in the transversality case of ∂*D* and *S*, using a Taylor series of the function (of two real variables) and a Maclaurin's expansion of the function (of one real variable) , we obtain an asymptotic expansion of *Ψ*(*z*) near *z*=0, of the form
Now, applying the Abel summability [1], ch. 4, we obtain the following asymptotic expansion.

The calculations made earlier and theorem 3.2 allow us to state the following theorem giving an asymptotic expansion of the integral *I*_{δ}(*λ*), in the situation where the surface *S* cuts tangentially the boundary ∂*D* of the domain *D*.

### Theorem 4.2

*We assume that the intersection of S and ∂D coincides with ∂S, ∂D and S are tangent at every point of their intersection ∂S, and the order of contact between S and ∂D is constant along ∂S (=p−1).*

*If condition (1.1) is fulfilled, δ is small enough and the function g is chosen so that condition (2.16) is fulfilled, then the following asymptotic expansion holds*.
*where*

(a)

*c*_{0}*and c*_{n}*(n even) are the same as in theorem*4.1.(b)

*for odd integers n*.(c)

*c*_{n}*and**represent the contribution of the points in the interior and the points in the boundary ∂S of the surface S, respectively*.

### Remark 4.3

In the tangency case of *S* and ∂*D*, the (*p*−1) terms in the expansion, of order of *λ*^{−(1/(p+1))1/r}, *λ*^{−(2/p+1)1/r},…,*λ*^{−((p−1)/p+1)1/r}, following the leading term *c*_{0}*λ*^{−1/r}, come entirely from the points of the boundary ∂*S* of the surface *S*, whereas in the transversality case, the points in the interior of the surface *S*, contribute in all terms of the expansion.

## 5. Influence of the geometry of *S*

To avoid the edge problem, we assume that *S* is a surface without boundary. We have from (3.3) and (3.4) (see theorem 3.2),
where
with
5.1The two last integrals in (5.1) are respectively equal to
and
So, we have

### Theorem 5.1

(1)

*If S is a surface without boundary, then**where b*_{n}=A_{n}+B_{n}+C_{n}, with(2)

*A*._{n}, n≥0 are the coefficients of the asymptotic expansion of I_{δ}(λ), in the case where S is a plane (H=K=0). The coefficients (B_{n}) and (C_{n}) characterize the contribution of the geometry of the surface S. More precisely, they characterize the contribution of the mean curvature H and the Gaussian curvature K of the surface S, respectively(3)

*B*._{0}=C_{0}=0. This means that the leading term in the expansion does not depend on the geometry of the surface S

## 6. Case when condition (1.1) is not satisfied

Consider new coordinates (*u*,*t*) defined by the transformations *M*_{l} given by (2.5). We have *F*_{l}(*u*,*t*)=*f*(*x*(*u*,*t*)). Define on *S* the function
We have assumed in this paper that the function is constant over *S*. Our approach fails in the vicinity of points of *S* where this function is not constant. Discontinuity of the function means that the order of degeneracy of the derivative of the function *f*, normal to the surface *S*, is not constant on *S*. Our problem reduces in this case, as we have already mentioned in the two-dimensional case in a previous paper [6], to the exploration of the contribution of points of discontinuity of the function *r*(*x*).

Here, we propose a method to deal with the problem in this situation. Indeed, assume that the function is discontinuous only on a finite number of points *x*_{1},*x*_{2},…,*x*_{k} of *S*. Let *U* be a neighbourhood of the set {*x*_{1},*x*_{2},…,*x*_{k}}. The technique used in this paper allows us to obtain asymptotic expansion of our integral on the set *D*_{δ}−*U*. About the expansion of our integral over a neighbourhood of each point *x*_{i}, *l*=1,…,*k*, we should use another kind of uniform asymptotic expansion which leads to special functions [14–17].

Consider, for example, the function
as a function of the real variable *t*, with two real parameters *u*_{1},*u*_{2}. If *r*(*x*_{i})=3, and *r*(*x*)=2 for *x*≠*x*_{i}, then we have
Thus, we claim that we may use the method described in Bleistein [14] to obtain a uniform asymptotic expansion of our integral with respect to parameters *u*_{1} and *u*_{2}.

## Acknowledgements

The authors thank referees for valuable advice.

- Received February 15, 2013.
- Accepted April 15, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.