Asymptotic analysis of Dotsenko-Fateev integrals

We develop a method for evaluating asymptotics of certain contour integrals that appear in Conformal Field Theory under the name of Dotsenko-Fateev integrals and which are natural generalizations of the classical hypergeometric functions. We illustrate the method by establishing a number of estimates that are useful in the context of martingale observables for multiple Schramm-Loewner evolution processes.


Introduction
In this paper, we will be interested in integrals of the form (z 1 +,z 2 +,z 1 −,z 2 −) A N j=1 (u − z j ) a j du, (1.1) where N 2 is an integer, {a j } N j=1 ⊂ R is a set of exponents, {z j } N j=1 ⊂ C is a finite collection of points, and the Pochhammer integration contour encloses two of these points, say z 1 and z 2 . More precisely, the Pochhammer contour in (1.1) begins at the base point A ∈ C, encircles z 1 once in the positive (counterclockwise) direction, returns to A, encircles z 2 once in the positive direction, returns to A, encircles z 1 once in the negative direction, returns to A, and finally encircles z 2 once in the negative direction before returning to A, see Figure 1. The remaining points {z j } N j=3 are assumed to lie exterior to the contour and the powers in the integrand are assumed to take their principal values at the starting point and are then analytically continued along the contour.
By employing what is sometimes called the screening method, Dotsenko and Fateev obtained representations of correlation functions in Conformal Field Theory (CFT) in terms of such integrals [3,4] (see also [2]). Integrals of the form (1.1) are therefore often referred to as Dotsenko-Fateev integrals. Various aspects of such integrals and related special functions have been studied in a number of places, e.g., [7,8]. For N = 2 and N = 3, the integral (1.1) can be expressed in terms of Gamma functions and the classical hypergeometric function 2 F 1 , respectively. In the case of N = 4, a comparison with the representation of Appell's F 1 function as an Euler integral (cf. [7]) shows that (1.1) can be viewed as an analytic continuation of Appell's function. We also mention [5,9,10] which discuss related questions in the context of the Schramm-Loewner Evolution (SLE).
The basic problem we are interested in here is how to compute the asymptotics of the integral (1.1) as one (or several) of the points z k , k = 3, . . . , N , approaches z 1 or z 2 . The computation of such asymptotics presents difficulties, because if the point z k , k = 3, . . . , N , approaches z 1 , say, then the contour gets squeezed between z k and z 1 which, in general, gives rise to a singular behavior. For N = 3, the relevant asymptotics can be obtained from well-known identities and expansions of hypergeometric functions. However, for N 4, the analysis is more intricate.
In this paper, we propose a method which makes it possible to compute the asymptotics of (1.1) to all orders for any N 3 as one (or many) of the points z k , k 3, approaches z 1 or z 2 . The obtained expansions are power series in the relevant small quantities with coefficients explicitly given in terms of Gamma functions and integrals of the form (1.1) of a lower order N −1. In the case of N = 4, it is conceivable that the asymptotic formulas we derive can be obtained also from known properties of Appell's function. However, as far as we are aware, formulas of this type have not appeared in the literature even in this simplest case.
By applying a linear fractional transformation, we may assume that z 1 = 0 and z 2 = 1 in (1.1); this yields an expression of the form (0+,1+,0−,1−) (1.2) For simplicity, we will present the results in this paper for the class of integrals corresponding to N = 4. Although the proposed method works for arbitrary values of N and an arbitrary number M N of merging points, its application involves a large number of steps when M is large (the number of steps typically grows like M , because, roughly speaking, each time two points merge one of the series contributing to the asymptotics splits into two).
1.1. Brief description of method. Let F (w 1 , w 2 ) ≡ F (a, b, c, d; w 1 , w 2 ) denote the integral in (1.2) for N = 4, that is, F (w 1 , w 2 ) is defined for w 1 , w 2 ∈ C \ [0, ∞) by where A ∈ (0, 1) is a base point, a, b, c, d ∈ R are real exponents, and w 1 , w 2 are assumed to lie outside the contour. In order to make F single-valued, we have restricted the domain of definition in (1.3) to w 1 , w 2 ∈ C \ [0, ∞). We will assume that a, d ∈ R \ Z are not integers, because otherwise the integral in (1.3) vanishes identically or can be computed by a residue calculation. Note that F can be analytically continued to a multiple-valued analytic function of w 1 , w 2 ∈ C \ {0, 1}. Our goal is to compute the asymptotic behavior of F to all orders as one or both of the points w 1 and w 2 approach 0 or 1. The basic idea is the following: If we want to consider the limit w 1 → 0 say, then we rewrite F as a sum of two terms (see equation (4.13)). One term which is defined by the same integral as F except that w 1 is now assumed to lie inside the contour in the same component as 0 (see equation (4.11)), and a second term which is defined by a similar expression but with the Pochhammer contour enclosing {0, w 1 } instead of {0, 1} (see (4.12) and (4.15)). The asymptotics of both of these terms can easily be computed to all orders by simply replacing the factors in the integrands by their asymptotic expansions as w 1 → 0 such as We emphasize that it is, in general, not possible to compute the asymptotics of F as w 1 → 0 by substituting the expansion (1.4) directly into (1.3). Indeed, such a procedure gives the correct contribution from the first term, but completely ignores the contribution from the second term.

Two examples.
Our initial motivation for studying asymptotics of integrals of the form (1.1) was that they appear naturally when constructing observables for multiple SLE curves via the screening method within CFT. Thus, in the second part of the paper, in order to illustrate our method, we consider two concrete examples of such observables where integrals of the form (1.1) are important. By applying the techniques developed in the first part of the paper, we are able to derive asymptotic estimates for the integrals in these examples. The estimates we establish are used in the derivation of the Green's function and Schramm's formula for multiple SLE presented in [11]. We expect similar asymptotic estimates to be needed also in the context of other SLE observables derived via the screening method.
1.3. Organization of the paper. In Section 2, we present four different examples of asymptotic expansions to all orders which can be derived by our method. Before turning to the full description of the method and the proofs of the above expansions in Section 4, we consider the hypergeometric case of N = 3 in Section 3 as motivation. The two examples from SLE theory are introduced in Section 5. In Section 6, we derive the estimates relevant for the first example corresponding to the Green's function. In Section 7, we derive the estimates relevant for the second example corresponding to Schramm's formula.

Four asymptotic theorems
The purpose of this paper is to propose a method which makes it possible to compute the asymptotics of (1.2) as one (or several) of the points w k , k = 3, . . . , N , approaches 0 or 1. There are clearly many different cases to consider depending on the value of N , on the number of points w k involved in the limiting process, and on whether each w k approaches 0 or 1. For the sake of presentation, we have chosen to discuss four cases in detail. The presented cases all have N = 4 and correspond to the following limits: (a) w 2 → 1, (b) w 1 → 0, (c) w 1 → 0 and w 2 → 0, and (d) w 1 → 0 and w 2 → 1. These four cases, which are treated one by one in Theorem 2.1-2.4, illustrate the different situations that may arise and it will be clear from the analysis of these cases how to apply the method also in other cases.
2.1. Notation. We let F (w 1 , w 2 ) ≡ F (a, b, c, d; w 1 , w 2 ), G(a, c, d; w), and H(a, d) denote the integral given in (1.2) for N = 4, N = 3, and N = 2, respectively. That is, where w is assumed to lie outside the contour, and H(a, d) is defined by The Pochhammer integration contour in (2.1) begins at the base point A ∈ (0, 1), encircles the point 0 once in the positive (counterclockwise) direction, returns to A, encircles 1 once in the positive direction, returns to A, and so on. The point w lies exterior to all loops; the factors in the integrand take their principal values at the starting point and are then analytically continued along the contour. A similar interpretation of Pochhammer contours applies to the definition (1.3) of F (w 1 , w 2 ) and elsewhere. Throughout the paper we adopt the convention that unless stated otherwise, the principal branch is used for all complex powers and logarithms, i.e., ln w = ln |w| + i arg w with arg w ∈ (−π, π]. For c ∈ C, we define ρ c (w) by .
Then F satisfies the following asymptotic expansion to all orders as Theorem 2.3 (Asymptotics as w 1 → 0 and w 2 → 0 with |w 1 /w 2 | < 1−δ). Let a, b, c, d ∈ R\Z be such that a+b, a+b+c / ∈ Z. Then F satisfies the following asymptotic expansion to all orders as w 1 → 0 and w 2 → 0 with w 1 , w 2 ∈ C \ [0, ∞) such that |w 1 /w 2 | < 1 − δ for some δ > 0: where the coefficients C kl (a, b, c, d), j = 1, 2, 3, are given by Theorem 2.4 (Asymptotics as w 1 → 0 and w 2 → 1). Let a, b, c, d ∈ R \ Z be such that a + b, c + d / ∈ Z. Then F satisfies the following asymptotic expansion to all orders as kl (a, b, c, d), j = 1, 2, 3, are given by Remark 2.5. For definiteness, we have stated Theorem 2.3 under the assumption that |w 1 /w 2 | < 1 − δ. It will be clear from the full description of the method in Section 4 that the case |w 2 /w 1 | < 1 − δ can be treated similarly. The case |w 1 | |w 2 | can also be handled by similar steps, but in this case the coefficients depend on the quotient α := w 2 /w 1 . In fact, a slight modification of the proof of Theorem 2.3 yields the following result (see Remark 4.4): If a, b, c, d ∈ R \ Z satisfy a + b, a + b + c / ∈ Z and w 2 = αw 1 , then F satisfies the following asymptotic expansion to all orders as w 1 → 0 and w 2 → 0 with w 1 , w 2 ∈ C \ [0, ∞) such that α ∈ C \ [0, ∞) and δ < |α| < δ −1 for some δ > 0: kl and the coefficients E Here G inside (a, b, c; w) is defined by the same formula (2.1) as G(a, b, c; w) except that w is assumed to lie inside the contour in the same component as 0. , where w lies exterior to the contour, then G andG are related by The expression (2.5) follows because the Pochhammer integral expression for the hypergeometric function 2 F 1 (see e.g., [12, Eq. (15.6.5)]) implies that , 3. The hypergeometric case of N = 3 Before turning to the full description of the method and the proofs of Theorem 2.1-2.4, it is helpful to consider, as motivation, the case N = 3 in which the integral in (1.2) reduces to a hypergeometric function.
Let G(a, c, d; w) be the function defined in (2.1) and corresponding to (1.2) with N = 3. Equation (2.5) expresses G(a, c, d; w) in terms of the hypergeometric function 2 F 1 and we can use known properties of this function to derive the asymptotics of G as w → 0 or w → 1. For definiteness, we consider the limit w → 1. We will show the following analog of Theorem 2.1. Proposition 3.1 (Asymptotics of G as w → 1). Let a, c, d ∈ R\Z be such that c+d / ∈ Z. Then G satisfies the following asymptotic expansion to all orders as w → 1 with w ∈ C \ [0, ∞): where the coefficientsÂ k (a, c, d), j = 1, 2, are given bŷ Proof. The function 2 F 1 (a, c, d; z) is an analytic function of z with a branch cut along [1, ∞); in particular, it is not analytic at z = 1. In order to find the asymptotics of G as w → 1, we therefore first use the hypergeometric identity (see [13,Eq. (10.12)]) The hypergeometric functions in (3.4) are analytic at w = 1. Hence we can write (3.4) asG whereP j (w) ≡P j (a, c, d; w), j = 1, 2, are analytic at w = 1. Recalling (2.6), it follows that G admits an expansion of the form (3.1) for some complex coefficientsÂ which are valid as w → 1, together with the identity where the Pochhammer symbol (a) k is defined by (a) k = Γ(a + k) Γ(a) = a(a + 1)(a + 2) · · · (a + k − 1).
However, in order to arrive at the simple expressions in (3.2), this approach requires a somewhat elaborate resummation of the coefficients and it is actually more convenient to derive (3.2) by proceeding as in the proof of Theorem 2.1 below.

Description of method
In this section, we describe our method by considering, in turn, the following four asymptotic sectors for the function F (w 1 , w 2 ) defined in (1.3): (a) w 2 near 1, (b) w 1 near 0, (c) w 1 and w 2 both near 0, and (d) w 1 near 0 and w 2 near 1. The limits considered in Theorem 2.1-2.4 belong to these four sectors, respectively, and the proofs of these theorems will also be given.
The basic idea of our method is to show that it is possible, for each asymptotic sector under consideration, to derive a generalization of the hypergeometric identity (3.5) from which asymptotics to arbitrary order can be obtained by simply replacing the factors in the integrand with their asymptotic expansions. Actually, we will see in Section 6 that it is often convenient in applications to work with the generalizations of the hypergeometric identity (3.5) themselves. These generalizations are presented in Proposition 4.1-4.5, respectively. Throughout the discussion, b, c ∈ R and a, d ∈ R \ Z denote some given parameters and we write F (w 1 , w 2 ) for F (a, b, c, d; w 1 , w 2 ). Furthermore, we let D 0 ⊂ C 2 and D 1 ⊂ C 2 denote the domains In the definition of the function P 1 , the point w 1 lies exterior to the contour, whereas w 2 lies inside the contour in the same component as 1. and where γ (w j ,∞) ⊂ C denotes a branch cut from w j to ∞. These branch cuts will be needed to make certain functions below single-valued; to be specific, we henceforth choose 4.1. The sector w 2 → 1. We will determine the behavior of F (w 1 , w 2 ) for w 2 close to 1 by deriving a generalization of (3.5).
where w 1 and w 2 lie exterior to the contour. Then where ρ c is the function in (2.3).
Assuming that c + d / ∈ Z, we define two functions P j : D 1 → C, j = 1, 2, as follows. The function P 1 is defined (up to a constant) by the same formula asF except that the point w 2 is assumed to lie inside the contour in the same component as 1; more precisely, for (w 1 , w 2 ) ∈ D 1 , where w 1 lies outside the contour and w 2 lies inside the contour in the same component as 1, see Figure 2. The function P 2 : D 1 → C is defined as follows. First, given for Re w 2 ∈ (0, 1) with Im w 2 > 0 sufficiently small by where A ∈ (0, 1) and the points w 2 w 2 −1 and w 1 −w 2 1−w 2 are assumed to lie exterior to the contour. Then, for each w 1 ∈ C \ [0, ∞), we use analytic continuation to extend P 2 to a (single-valued) analytic function of w 2 ∈ C \ ((−∞, 1] ∪ γ (w 1 ,∞) ). The latter step is permissible because the function P 2 can be analytically continued as long as the points w 2 w 2 −1 and w 1 −w 2 1−w 2 stay away from the set {0, 1, ∞}, i.e., as long as w 2 / ∈ {0, 1, w 1 , ∞}. Let f (w 1 , w 2 ±i0) denote the boundary values of a function f (w 1 , w 2 ) as w 2 approaches the real axis from above and below, respectively. The following lemma provides the desired generalization of the hypergeometric identity (3.5).
Then the function F obeys the identity It is actually enough to show that Indeed, for each w 1 ∈ C \ [0, ∞), both sides of the equation (4.7) are analytic functions of w 2 ∈ C\((−∞, 1]∪γ (w 1 ,∞) )} which can be extended to multiple-valued analytic functions of w 2 ∈ C \ {0, 1, w 1 }. Hence (4.7) follows from (4.8) by analytic continuation. Let us prove (4.8). Let > 0 be small. Let w 1 ∈ C \ [0, ∞) and w 2 ∈ (0, 1). Given w ∈ C, let , denote counterclockwise semicircles of radius centered at w, see Figure 3. Furthermore, oriented so that 4 1 L j A is a counterclockwise contour enclosing 0 and 1, see Figure 4. Then we can writẽ Simplification gives Hencẽ Using the identity we can write this as That is, Performing the change of variables s = v−w 2 1−w 2 , which maps the interval (w 2 , 1) to the interval (0, 1), this yields Comparing this expression with the definition (4.5) of P 2 , equation (4.8) follows.
Since both terms on the right-hand side of the identity (4.6) are well-behaved for w 2 close to 1, the behavior of F as w 2 → 1 can easily be extracted from this identity.
Proof of Theorem 2.1. Suppose a, b, c, d ∈ R \ Z and c + d / ∈ Z. Using the identity (4.6), we can easily prove Theorem 2.1. Indeed, the functions P 1 and P 2 in (4.6) admit asymptotic expansions to all orders as follows. Substituting the expansion into the definition of P 1 (w 1 , w 2 ) and recalling that w 2 and 1 lie in the same component inside the contour, we find where the integral on the right-hand side is exactly G(a, b, c + d − k; w 1 ). Similarly, substituting the expansions into the definition (4.5) of P 2 (w 1 , w 2 ), we find, as w 2 → 1, Substituting (4.9) and (4.10) (with the summation variable m replaced by k = m + l) into (4.6), we arrive at the expansion given in Theorem 2.1.

4.2.
The sector w 1 → 0. In order to determine the behavior of F (w 1 , w 2 ) as w 1 → 0, we define two functions Q j : D 0 → C, j = 1, 2, as follows. The function where A ∈ (0, 1), w 1 lies inside the contour in the same component as 0, and w 2 lies outside the contour. Given where A ∈ (0, 1) and the points w 2 w 1 and 1 w 1 lie exterior to the contour. For each w 2 ∈ C \ [0, ∞), we then use analytic continuation to extend Q 2 to a function of We have the following analog of Proposition 4.1.
Proof. By analyticity, is enough to show that for w 1 ∈ (0, 1) and where w 2 lies exterior to the contours. Moreover, Simplification gives Hence Using the identity That is, Performing the change of variables s = v w 1 , which maps the interval (0, w 1 ) to the interval (0, 1), we obtain The lemma follows.
Proof of Theorem 2.2. Suppose a, b, c, d ∈ R\Z and a+b / ∈ Z. In the same way that (4.6) can be used to determine the asymptotics of F as w 2 → 1, the identity (4.13) can be used to determine the asymptotics of F (w 1 , w 2 ) as w 1 → 0. Indeed, the expansion given in Theorem 2.2 follows from (4.13) after substituting the following asymptotic expansions as w 1 → 0 into the definitions of Q 1 and Q 2 : 4.3. The sector w 1 → 0 and w 2 → 0. We next determine the asymptotics of F (w 1 , w 2 ) in the regime where both w 1 and w 2 approach zero. Assuming a + b, a + b + c / ∈ Z, we define two functions R j : D 0 → C, j = 1, 2, as follows. The function where A ∈ (0, 1) and both points w 1 and w 2 are assumed to lie inside the contour in the same component as 0. For 0 < Re w 1 < Re w 2 < 1 with Im w 1 < 0 and Im w 2 < 0, we define R 2 (w 1 , w 2 ) by where we assume A ∈ (0, 1) is so large that Re (Aw 2 − w 1 ) > 0, that the point w 1 w 2 lies inside the contour in the same component as 0, and that 1 w 2 lies outside the contour. We then use analytic continuation to extend R 2 to all of D 0 . We have the following analog of Proposition 4.1. a, b, c, d ∈ R and a, d, a + b, a +

Proposition 4.3. Suppose
Then the function F obeys the following identity for (w 1 , w 2 ) ∈ D 0 : Proof. Both sides of (4.18) are analytic functions of (w 1 , w 2 ) ∈ D 0 which extend to multiple-valued analytic functions of Hence, by Proposition 4.2, it is enough to show that where, for a function f , we use the short-hand notation f − (w 1 , w 2 ) := f (w 1 −i0, w 2 −i0).
Let 0 < w 1 < w 2 < 1 and suppose 0 < < 1 2 min{w 1 , w 2 − w 1 , 1 − w 2 }. Then where the principal branch is used for all powers. A computation gives Using the identity That is, where w 1 lies inside the contour in the same component as 0. Applying the change of variables s = v w 2 , which maps the interval (0, w 2 ) to the interval (0, 1), we obtain where A ∈ (0, 1) is so large that Aw 2 − w 1 > 0. Equation (4.19) follows.
Proof of Theorem 2.3. Suppose a, b, c, d ∈ R \ Z and a + b, a + b + c / ∈ Z. Theorem 2.3 follows by expanding the integrands in the definitions of R 1 , R 2 , Q 2 as w 1 → 0 and w 2 → 0, and substituting the resulting expressions into (4.18). We have stated Theorem 2.3 under the assumption that |w 1 /w 2 | < 1 − δ; hence we use the expansion (4.16b) of (sw 1 − w 2 ) c and the expansion of the factor (sw 2 − w 1 ) b .

Remark 4.4.
To derive the expansion (2.4) of F as w 1 , w 2 → 0 with |w 1 | |w 2 |, we proceed in the same way as the proof of Theorem 2.3, i.e., we expand R 1 , R 2 , Q 2 as w 1 , w 2 → 0 and substitute the resulting expressions into (4.18). However, in this case, since |w 1 /w 2 | is not necessarily smaller than 1, we do not use the expansions (4.16b) and (4.20) of (sw 1 − w 2 ) c and (sw 2 − w 1 ) b ; instead we simplify the expressions for Q 2 and R 2 using the identities sw 1 − w 2 = w 1 (s − α) and sw 2 − w 1 = w 2 (s − α −1 ) where α = w 2 /w 1 ; then the remaining factors are expanded as in the proof of Theorem 2.3.

4.4.
The sector w 1 → 0 and w 2 → 1. We finally consider the behavior of F (w 1 , w 2 ) when w 1 is near 0 and w 2 is near 1. Assuming that a + b, c + d / ∈ Z, we define two functionsQ 1 : D 1 → C and T 1 : D 0 → C as follows. We defineQ 1 for (w 1 , w 2 ) ∈ D 1 bỹ where A ∈ (0, 1), w 1 lies inside the contour in the same component as 0, and w 2 lies outside the contour. Then where ρ c is given by (2.3). For 0 < Re w 1 < Re w 2 < 1 with Im w 1 < 0 and Im w 2 > 0, we define T 1 (w 1 , w 2 ) by where w 1 lies inside the contour in the same component as 0, w 2 lies inside the contour in the same component as 1, and we assume that Re w 1 < A < Re w 2 . We then use analytic continuation to extend T 1 to all of D 1 . a, b, c, d ∈ R and a, d, a +
Proof of Theorem 2.4. By expanding the integrands in the definitions of T 1 , P 2 , Q 2 as w 1 → 0 and w 2 → 1, and substituting the resulting expressions into (4.21), Theorem 2.4 is obtained after a lengthy computation.

Two examples from SLE theory
In this section, in an effort to illustrate the method described above, we present two examples from SLE theory which involve Dotsenko-Fateev integrals of the form (1.1). The first example is related to the Green's function observable for two commuting SLE curves and involves an integral of the form (1.1) with N = 4 and (see (5.2)) where α > 1 is a parameter. The second example is related to Schramm's formula for the same SLE system and involves an integral of the form (1.1) with N = 4 and (see (5.12)) a 1 = α, For each example, we derive the asymptotic estimates which are needed in order to establish that the relevant integral describes the given observable.

Multiple SLE and Dotsenko-Fateev integrals.
Let us briefly recall the definition of multiple SLE systems and describe how Dotsenko-Fateev integrals arise when constructing observables for such systems via the screening method, see [11] for a more complete discussion. SLE κ curves are constructed by solving Loewner's differential equation where the driving term ξ 1 t is standard Brownian motion. The curve itself is defined by lim y→0+ g −1 t (ξ 1 t + iy); this is a random continuous curve growing from 0 to ∞ in the upper half-plane H = {Im z > 0}. If κ 4, the curve is simple and stays in H for t > 0. SLE curves appear as scaling limits of interfaces in various critical lattice models. It is natural to consider scaling limits of multiple interfaces simultaneously, and this leads to multiple SLE. We are interested in multiple SLE with two curves started from ξ 1 , ξ 2 ∈ R, respectively, and growing towards ∞ in H, see [6]. The marginal law of the SLE started from ξ 1 is that of a variant of SLE with an additional marked boundary point at ξ 2 . For this variant, which is called SLE κ (2), the dynamics of the driving term is given by the system dξ 1 t = dB t + (2/κ)/(ξ 1 t − ξ 2 t ), where B t is standard Brownian motion, ξ 2 t := g t (ξ 2 ), and g t solves (5.1) with ξ 1 t as driving term. An important feature is that the system can be grown in a commutative way [6]. The extra drift term entails serious difficulties when constructing observables for such SLE systems.
In [11], the screening method is used to derive explicit formulas for two of the most natural SLE observables: the renormalized probability that the system passes infinitesimally near a given point in H (the Green's function) and the probability that the system passes to the right of a given point in H (Schramm's formula). The derivation in [11] proceeds as follows: First, using a CFT description of the multiple SLE system, the screening method is employed to generate explicit "guesses" for the given observables in terms of Dotsenko-Fateev integrals. The "guesses" are then shown to indeed describe the desired probabilities via a sequence of probabilistic arguments. The latter arguments rely heavily on appropriate asymptotic estimates for the relevant Dotsenko-Fateev integrals. In the remainder of this paper, we derive the estimates needed in [11].

2)
where the constantĉ ≡ĉ(κ) is given bŷ This definition of G(z, ξ 1 , ξ 2 ) can be extended to all α > 1 by continuity, see [11]. It follows from (5.2) and (5.3) that the product y 1− 1 α G(z, ξ 1 , ξ 2 ) only depends on the two angles θ 1 and θ 2 defined by θ j := arg(z − ξ j ), j = 1, 2, see [11, Section 6.2.1]. Hence we may define the function h by By applying the method of Section 2-4, we can prove the following proposition which is used in [11] to derive a formula for the Green's function.

Proposition 5.1 (Estimates for Green's function). Let α
2. Then the function h(θ 1 , θ 2 ) defined in (5.5) is a smooth function of (θ 1 , θ 2 ) ∈ ∆ and has a continuous extension to the closure∆ of ∆. This extension satisfies where h f (θ) is defined by Im z Figure 6. The integration contour used in the definition (5.11) of M(z, ξ) is a path fromz to z which passes to the right of ξ.
Moreover, there exists a constant C > 0 such that Proof. See Section 6.

Example 2: Schramm's formula.
Let α > 1 and define the function M(z, ξ) by (5.11) where J(z, ξ) is the integral defined by with the integration contour fromz to z passing to the right of ξ, see Figure 6. Moreover, define the function P (z, ξ) by where the normalization constant c α ∈ R is given by . (5.14) We will prove the following proposition which establishes the properties of P needed for the proofs in [11].

Proof of Proposition 5.1
By applying the method developed in Section 2-4, we can determine the behavior of the Dotsenko-Fateev integral in (5.2). This will lead to asymptotic formulas for the behavior of h(θ 1 , θ 2 ) near the boundary of ∆ from which Proposition 5.1 will follow.
Let F (a, b, c, d; w 1 , w 2 ) be the function defined in (1.3) with a, b, c, d given by i.e., for w 1 , w 2 ∈ D 0 , where A ∈ (0, 1) is a basepoint and w 1 , w 2 are assumed to lie outside the contour. The Pochhammer contour in (5.2) encloses the variable points z and ξ 2 . In order to easily apply the results from Section 2-4, we first need to express h in terms of the integral F whose contour encloses the fixed points 0 and 1. This can be achieved by applying a linear fractional transformation which maps z and ξ 2 to 0 and 1, respectively.
Assume that α 2 is such that 3α 2 , 2α / ∈ Z; the cases when 3α 2 and/or 2α is an integer will be considered separately . Then a, b, c, d ∈ R \ Z and a + b, c + d, a + In fact, since a + b = 2α − 2 0, it can be seen from Lemma 6.4 and Theorem 2.2 that the function F (w 1 , w 2 ) in (6.3) is bounded in the sector S 2 . On the other hand, since c + d + 1 = 1 − α < −1, the sum in Theorem 2.1 which involves the coefficients A (2) k is, in general, singular as w 2 → 1. However, it turns out that the contribution from this sum to h vanishes identically because of the taking of the imaginary part in (6.3). In order to see this, we need to consider the function P 2 of Proposition 4.1 from which the coefficients A (2) k originated in more detail. Lemma 6.5. Let P 2 (w 1 , w 2 ) denote the function defined in (4.5) with a, b, c, d given by (6.1) and define X : ∆ → C by where the variables w 1 and w 2 are given by (6.4). Then Proof. From the definition (4.5) of P 2 we see that for w 1 ∈ C \ [0, ∞) and w 2 ∈ (0, 1) we have By definition, the value of P 2 (w 1 , w 2 ) at a general point (w 1 , w 2 ) ∈ D 1 is determined by analytic continuation of (6.10) within the connected set D 1 ⊂ C 2 . The branches of the complex powers in (6.10) are fixed by requiring that the principal branch is used initially at the basepoint s = A ∈ (0, 1); for definiteness, let us choose A = 1/2. This means that whenever the points cross the negative real axis during the analytic continuation, extra factors of e ±2πia and e ±2πib , respectively, have to be inserted in (6.10).
In order to evaluate the function X in (6.8), we need the value of P 2 at points (w 1 , w 2 ) ∈ E, where E denotes the subset of C 2 characterized by (6.4), i.e., E = (w 1 , w 2 ) = 1 − e −2iθ 2 , sin θ 2 sin θ 1 e −i(θ 2 −θ 1 ) (θ 1 , θ 2 ) ⊂ ∆ . If w 1 and w 2 are given by (6.4), then cot θ 2 < cot θ 1 , 1 Hence, we have, for all (w 1 , w 2 ) ∈ E, This shows that neither of the points in (6.11) crosses the negative real axis as long as (w 1 , w 2 ) remains within E. We can therefore find a formula for P 2 valid in E as follows. Let (w 1 , w 2 ) be a point in E corresponding to (θ 1 , θ 2 ) via (6.4). Then Let 0 < < |w 1 − 1| be small and let (w 1 (t),w 2 (t)), t ∈ [0, 1], be the path in D 1 defined byw 1 (t) = w 1 for all t, while the pathw 2 (t) starts at 1 − + i0, proceeds clockwise around the small circle of radius centered at 1 until it reaches the point 1 + e −iθ 2 , and then proceeds along the straight line segment [1 + e −iθ 2 , w 2 ] until it reaches w 2 . Asw 2 moves along the arc from 1 − + i0 to 1 + e −iθ 2 , the point A − 1 − 1 w 2 (t)−1 crosses the negative real axis from the upper into the lower half-plane once (this adds a factor of e 2πia to (6.10)), and, provided that Im w 1 0 (i.e. θ 2 π/2), A − 1 +w 1 (t)−1 w 2 (t)−1 also crosses the negative real axis from the upper into the lower half-plane once (this adds a factor of e 2πib to (6.10)). If Im w 1 > 0, then A − 1 +w 1 (t)−1 w 2 (t)−1 does not cross the negative real axis. By varying θ 1 in (6.13), we see that the part of the path for whichw 2 belongs to the segment [1 + e −iθ 2 , w 2 ] lies in E; hence the analytic continuation along this part adds no more factors to (6.10). We end up with the following formula for P 2 on E: where 1 + 1 w 2 −1 and 1 − w 1 −1 w 2 −1 lie exterior to the contour. Substituting this formula into (6.8) and simplifying, we find where w 1 , w 2 are given by (6.4). But and, by (6.12), (6.14) If g(s) is an analytic function, then the general identity γ g(w)dw = γ g(v)dv implies Using this identity to compute the imaginary part of (6.14) we arrive at This completes the proof of the lemma.
In order to prove Proposition 5.1, we only need leading and subleading estimates on F , so we shall be content with this level of precision. The required bounds on the functions P 1 , Q j , R j , T 1 , F are then collected in the next lemma. Lemma 6.7. Suppose a, b 0 and c, d 0 satisfy a, d, a + Then the following estimates hold: (a) |P 1 (w 1 , w 2 )| C and |P 1 (w 1 , w 2 )−P 1 (w 1 , 1)| C|w 2 −1| uniformly for all (w 1 , w 2 ) ∈ C 2 such that dist(w 1 , {0, 1, ∞}) > and |w 2 − 1| < 1 − .
Proof. The estimates follow directly from the definitions of the functions P 1 , Q j , R j , T 1 , and F . Remark 6.8. If (w 1 , w 2 ) lies on a branch cut, the bounds in Lemma 6.7 should be interpreted as saying that both the left and right boundary values obey the bounds.
We are now in a position to prove Proposition 5.1. Indeed, since h clearly is smooth in the interior of ∆ and the parameter δ > 0 which defines the sectors S j is arbitrary, Proposition 5.1 follows from the following result. Lemma 6.9. Let α 2. Then the function h(θ 1 , θ 2 ) defined in (5.5) satisfies the following estimates: where h f (θ) is defined in (5.8).
Taking the definition (5.4) ofĉ into account, it follows that h(θ 1 , π) = sin α−1 θ 1 . This proves (6.21). Lemma 6.4 (4), and Lemma 6.7 (c) and (f ) show that Hence equation (6.18) implies that (6.19) holds in S 4 . Lemma 6.4 (5), and Lemma 6.7 (g) show that Hence equation (6.3) shows that (6.19) holds in S 5 . This completes the proof of the lemma in the case when 3α 2 and 2α are not integers. Assume now that 3α 2 and/or 2α is an integer. Then some of the functions in Lemma 6.7 degenerate, so a slightly different argument is required. We do not give complete details, but outline the relevant steps.
Suppose first that α / ∈ Z but 3α 2 or 2α is an integer. Then the limit w 2 → 1 can still be treated as before, because c + d = −α is not an integer. However, the limits involving w 1 → 0 or w 2 → 0 cannot be treated in the same way in general, because a + b = 2α − 2 and/or a + b + c = 3α 2 − 2 is an integer. However, since α 2, we have a + b > 0 and a + b + c > 0. Hence the integral (6.2) defining F is nonsingular at v = 0 (also in the limit as w 1 and w 2 approach zero). Hence, we can derive the leading behavior of F in these regimes using the following alternative approach: First, we collapse the two loops of the Pochhammer contour enclosing the origin down to the interval [0, A]. Then we find the leading-order asymptotics by Taylor expanding the integrand as w 1 and/or w 2 approaches zero.
Assume finally that α = n 2 is an integer. This case is considered in [11], where an expression for h(θ 1 , θ 2 ; n) is derived by taking the limit of the defining equation (6.2) for F as α → n. In order to prove (6.19)-(6.21) in this case, we compute the limits as α → n of each of the four equations in Lemma 6.6. This gives four analogous equations valid for α = n. As above, it follows from these equations that h satisfies (6.19)-(6.21). The crucial point is that the singular contribution from P 2 vanishes as a consequence of (6.9).
Proof. Since α > 1, the integral defining J(z, ξ) is convergent for each z ∈ H and each ξ > 0. To prove the smoothness of J, we first assume that α > 1 is an integer. In this case the integral in (5.12) can be computed explicitly in terms of logarithms and powers of z,z, z − ξ, andz − ξ (see [11] for the case α = 2). Hence J(z, ξ) is smooth for (z, ξ) ∈ H × (0, ∞). Assume α > 1 is not an integer. Then, fixing a basepoint A > ξ, we can rewrite the expression (5.12) for J(z, ξ) as where the integration contour is the composition of four loops {l j } 4 1 based at A (see Figure 11) and the integrand is evaluated using analytic continuation along the contour. More precisely, the loop l 1 encircles z once in the counterclockwise direction, l 2 encircles z once in the counterclockwise direction, l 3 encircles z once in the clockwise direction, and l 4 encirclesz once in the clockwise direction. On the first half of l 1 , the principal branch is used, but as the contour l 1 encircles z in the counterclockwise direction, the power (u − z) α in the integrand picks up an additional factor of e 2iπα with respect to the principal branch; then, as l 2 encirclesz in the counterclockwise direction, the power (u−z) α−2 in the integrand picks up the factor e 2iπ(α−2) and so on. Collapsing the contour onto a single path fromz to z and collecting the exponential factors, we see that (7.1) reduces to (5.12). Since the contour in (7.1) avoids the branch points, the integral in (7.1) can be differentiated an unlimited number of times with respect to z,z, and ξ. This completes the proof of the lemma.

Remark 7.2 (A main difference between the two examples). The integrals relevant for
Example 2 have better convergence properties than those of Example 1. Indeed, the Pochhammer integral expression (7.1) for the function J(z, ξ) relevant for Example 2 is well-defined for any α ∈ R \ Z. If α > 1 is not an integer, the expression (5.12) for J can be recovered from (7.1) by collapsing the contour down to a simple path fromz to z. The integral in (5.12) converges because the exponents of the factors (u − z) α and (u −z) α−2 are both > −1 when α > 1.
On the other hand, the Pochhammer contour appearing in the definition (5.2) of the function I(z, ξ 1 , ξ 2 ) relevant for Example 1 encircles the points z and ξ 2 and the corresponding integrand involves the factors (u − z) α−1 and (ξ 2 − u) −α/2 . Thus, this contour can only be collapsed down to simple path from z to ξ 2 if 0 < α < 2. Since Proposition 5.1 is stated under the assumption that α 2, this means that the analysis of Example 1 requires the use of a Pochhammer contour (in fact, the contour can be collapsed at z, but not at ξ 2 ). Remark 7.2 suggests that it should be easier to prove Proposition 5.2 than Proposition 5.1. This is indeed the case. It is of course still possible to proceed as in Example 1 and use the full machinery of Section 2-4 to prove Proposition 5.2. However, in order to complement the discussion of Example 1, we will in what follows present a more elementary proof of Proposition 5.2 which relies on direct estimates. The takeaway is that in some simpler cases one has the choice of either using the machinery of Section 2-4 or a more naive approach based on direct estimates. Lemma 7.3. The function J(z, ξ) defined in (5.12) satisfies the following estimates: Proof. To prove (7.2a), we let z = ξ + re iθ and choose the following parametrization of the integration contour in (5.12) (see Figure 12): This yields after simplification r > 0, θ ∈ (0, π), ξ > 0. (7.4) It follows that for all r > 0, ϕ ∈ (−π, π), and ξ > 0, we obtain the estimate The integral remains bounded as θ ↑ π, because 3α 2 − 2 > −1. Thus we arrive at |J(ξ + re iθ , ξ)| Cr α−1 , r > 0, θ ∈ (0, π), ξ > 0, which is (7.2a).
To prove (7.2b), we let z = x + iy in (5.12) and use the parametrization u = x + is, −y s y, of the contour fromz to z. Assuming that x > ξ, this yields Re z Im z Figure 12. The contour fromz = ξ + re −iθ to z = ξ + re iθ defined in equation (7.3).
It follows that Re z Im z Figure 13. The integration contour L x in (7.8) is a loop which encloses ξ in the counterclockwise direction.
Lemma 7.8 and Lemma 7.10 show that the right-hand side of (7.32) is independent of r > 0. We can therefore evaluate it in the limit as r ↓ 0. Recalling the definition of M, we write The function J is bounded on each compact subset ofH by Lemma 7.4. Hence f r (ϕ) obeys the estimate |f r (ϕ)| C sin α−2 ϕ, r < ξ/2, ϕ ∈ [0, π].