Adv. appl. Clifford alg. 20 (2010), 379–391 c 2009 Birkh¨ auser / Springer Basel AG, Switzerland 0188-7009/020379-13, published online August 6, 2009 DOI 10.1007/s00006-009-0179-y
Advances in Applied Clifford Algebras
A Note on the Fueter Theorem Dixan Pe˜ na Pe˜ na and Frank Sommen Abstract. In this paper we present the most general form of Fueter’s theorem obtained so far. Some examples of applications are also computed. Mathematics Subject Classification (2000). 30G35. Keywords. Clifford analysis, Fueter’s theorem.
1. Introduction Fueter’s theorem is named after the Swiss mathematician R. Fueter who in his 1935-paper [4] obtained a method to generate monogenic quaternionic functions starting from a holomorphic function in the upper half of the complex plane. More precisely, if f (z) = u(x, y) + iv(x, y) (z = x + iy) is a holomorphic function in some open subset Ξ ⊂ C+ = {z = x + iy ∈ C : y > 0}, then in the corresponding region, the function q F (q0 , q) = ∆ u(q0 , |q|) + v(q0 , |q|) |q| is both left and right monogenic with respect to the quaternionic Cauchy-Riemann operator D = ∂q0 + i∂q1 + j∂q2 + k∂q3 , i.e. DF = F D = 0. Here q = q1 i + q2 j + q3 k is a pure quaternion and ∆ = ∂q20 + ∂q21 + ∂q22 + ∂q23 denotes the Laplace operator in four dimensional space. Under the same assumption on f , Fueter’s theorem has been extended successively in a Clifford analysis setting by Sce [17], Qian [14] and Sommen [18] (see also [5, 6, 7]). In [16] (see also [8]) Qian and Sommen proposed a new generalization using functions satisfying the Riesz system (monogenic 1-vector functions) instead of the The first author was supported by a Doctoral Grant of the Special Research Fund of Ghent University.
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usual holomorphic functions. Recently, the authors proved a more general version with monogenic l-vector functions as initial functions (see [9]). In this paper our goal is to show that it is possible, in general starting from monogenic functions in a certain Clifford algebra, to generate monogenic functions in another Clifford algebra of higher dimension. In this way we present the most general form of Fueter’s theorem obtained so far. Apart from being remarkable and appealing in its own right, Fueter’s technique has played a key role in the study of Fourier multipliers and singular integrals on the unit sphere (see [15]). We should mention that it also provides us with some special monogenic functions (see e.g. [5, 16]). In the forthcoming paper [13], the authors have used this result to obtain a closed formula for the CK-extension of the Gaussian distribution in Rm and the Gaussian fundamental solution which generalizes the complex fundamental solution exp(z 2 /2)/z. The outline of the paper is as follows. In the next section, we briefly recall some of the basic notions of Clifford analysis. The last section contains the statement and the proof of our result together with some examples.
2. Clifford Analysis Clifford analysis may be regarded as a generalization to a higher dimensional setting of the theory of holomorphic functions in the complex plane. It involves the study of functions on Euclidean space with values in a Clifford algebra (see e.g. [1, 3]). We denote by R0,m (m ∈ N) the real Clifford algebra constructed over the orthonormal basis (e1 , . . . , em ) of the Euclidean space Rm . The basic axiom of this associative but non-commutative algebra is that the product of a vector mwith itself equals its squared length up to a minus sign, i.e. for any vector x = j=1 xj ej in Rm , we have that m x2j . x2 = −|x|2 = − j=1
It thus follows that the elements of the basis submit to the multiplication rules e2j = −1, ej ek + ek ej = 0,
j = 1, . . . , m, 1 ≤ j = k ≤ m.
A basis for the algebra R0,m is then given by elements of the form eA = ej1 · · · ejk , where A = {j1 , . . . , jk } ⊂ {1, . . . , m} is such that j1 < · · · < jk . For the empty set ∅, we put e∅ = e0 = 1, the latter being the identity element. It follows that the dimension of R0,m is 2m .
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Any Clifford number a ∈ R0,m may thus be written as aA eA , aA ∈ R, a= A
m
or still as a = k=0 [a]k , where [a]k = |A|=k aA eA is a so-called k-vector (k = 0, 1, . . . , m). In particular, the 0-vectors and 1-vectors are simply called scalars and vectors respectively. (k) Denoting the space of k-vectors by R0,m , we have that R0,m =
m
(k)
R0,m .
k=0
Observe that R may be naturally embedded in the Clifford algebra R0,m by associating to any element (x0 , x1 , . . . , xm ) ∈ Rm+1 the “paravector” x given by m+1
x = x0 + x. Conjugation in R0,m is defined as the anti-involution for which ej = −ej , j = 1, . . . , m. By means of the conjugation, a norm |a| may be defined for each a ∈ R0,m by putting a2A . |a|2 = [aa]0 = A
Clifford algebras (also called geometric algebras) were introduced in 1878 by the English geometer and philosopher W. K. Clifford, generalizing the complex numbers and Hamilton’s quaternions (see [2]). They have important applications in geometry and theoretical physics. The first-order differential operator ∂x = ∂x0 + ∂x = ∂x0 +
m
ej ∂xj ,
j=1
called generalized Cauchy-Riemann operator, factorizes the Laplace operator in Rm+1 : m ∂x2j = ∂x ∂ x = ∂ x ∂x . (2.1) ∆x = j=0
The main objects of study in Clifford analysis are the so-called monogenic functions which may be described as null solutions of the generalized Cauchy-Riemann operator ∂x . More precisely, a function f (x) defined and continuously differentiable in an open set Ω of Rm+1 and taking values in R0,m , is called a (left) monogenic function in Ω if and only if it fulfills in Ω the equation ∂x f ≡
m
ej eA ∂xj fA = 0.
j=0 A
From (2.1) it follows that any monogenic function in Ω is harmonic in Ω and hence real-analytic in Ω.
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For a differentiable k-vector valued function Fk = |A|=k Fk,A eA and a differentiable R0,m -valued function g, we have the following Leibniz rule (see e.g. [10, 12]) ∂x (Fk g) = (∂x Fk )g + 2
m
[ej Fk ]k−1 (∂xj g) + (−1)k Fk (∂x g).
(2.2)
j=1
3. Generalized Fueter’s Theorem
d Consider the decomposition Rm = s=1 Rps , where p1 , . . . , pd are positive integers d such that s=1 ps = m. For any x ∈ Rm , we may write x=
d
x(s) ,
x(s) =
s=1
ps
(s) (s)
xj ej
j=1
and accordingly ∂x =
d
∂x(s) ,
∂x(s) =
s=1
ps
(s)
ej ∂x(s)
j=1 (s)
j
(s)
where the meaning of the notations xj and ej is obvious. We will denote by R0,d the real Clifford algebra generated by the elements Es , s = 1, . . . , d, with the usual multiplication rules Es2 = −1, Es Es + Es Es = 0,
s = 1, . . . , d 1 ≤ s = s ≤ d.
In what follows, we will consider an arbitrary but fixed function G on Rd+1 with values in R0,d in the variables y0 , y1 , . . . , yd . Such a function can be written as GB (y0 , y1 , . . . , yd )EB G(y0 , y1 , . . . , yd ) = B
where EB = Eβ1 · · · Eβl and B = {β1 , . . . , βl } ⊂ {1, . . . , d} is such that β1 < · · · < βl . We will also assume that the function G is monogenic with respect to the generalized Cauchy-Riemann operator ∂y0 + ∂y = ∂y0 +
d
Es ∂ys
s=1
in some open subset Ξ ⊂ (y0 , y1 , . . . , yd ) ∈ Rd+1 : ys > 0, s = 1, . . . , d , i.e. (∂y0 + ∂y )G = 0,
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or equivalently, for each l = 0, . . . , d
d
|B|=l−1 s=1 s∈B /
∂ys GB Es EB +
∂y0 GB EB
|B|=l l+1
+
(−1)s ∂yβs GB EB\{βs } = 0. (3.1)
|B|=l+1 s=1
We prove the following generalization of Fueter’s theorem. Theorem 3.1. Let G be as above. Assume ps (s = 1, 2, . . . , d) to be odd and let Pk (x) =
d
Pks x(s)
s=1
with k = s=1 ks and Pks x(s) a homogeneous monogenic polynomial of degree ks in Rps with values in the real Clifford algebra constructed over Rps . Then k+ m−d 2 GB (x0 , r1 , . . . , rd ) ω B Pk (x) ∆x d
B
= {x ∈ Rm+1 : (x0 , r1 , . . . , rd ) ∈ Ξ}. Here, for any B = is monogenic in Ω {β1 , . . . , βl } ⊂ {1, . . . , d} with β1 < · · · < βl , we have put ω B = ω β1 · · · ωβl and ω ∅ = 1, where ωs = x(s) /rs , with rs = |x(s) |, s = 1, . . . , d. We have divided the proof into a series of lemmas. For the proof of Lemmas 3.2 and 3.3 we refer the reader to [9]. Lemma 3.2. Suppose that f (t1 , . . . , td ) and g(t1 , . . . , td ) are real-valued infinitely differentiable functions on Rd and that Dtj and Dtj are differential operators defined by Dtj (0){f } = Dtj (0){f } = f and for n ≥ 1
n 1 Dtj (n){f } = ∂t {f }, j = 1, . . . , d, tj j tj D (n − 1){f } tj D (n){f } = ∂tj , j = 1, . . . , d. tj Then one has (i) Dtj (n){∂tj f } = ∂tj Dtj (n){f }, (ii) Dtj (n){∂tj f } − ∂tj Dtj (n){f } = 2n/tj Dtj (n){f }.
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Lemma 3.3. Assume that h(x0 , r1 , . . . , rd ) is a real-valued harmonic function in the variables x0 , r1 , . . . , rd . Then
d d
n! n ∆x h ω B Pk (x) = dk ,p (ns ) Drs (ns ) n1 ! · · · nd ! s=1 s s s=1 s∈B /
×
l
D
rβc
(nβc ){h} ωB Pk (x)
c=1
where the summation runs over all possible n1 , . . . , nd ∈ N0 such that d
ns = n,
s=1
and dks ,ps (ns ) = (2ks + ps − 1)(2ks + ps − 3) · · · (2ks + ps − (2ns − 1)) dks ,ps (0) = 1. Lemma 3.4. Let AB (x0 , r1 , . . . , rd ) be a real-valued continuously differentiable functions in the variables x0 , r1 , . . . , rd . Then the function
AB (x0 , r1 , . . . , rd ) ω B Pk (x) B
is monogenic if for each l = 0, . . . , d d ∂rj AB ω j ω B + ∂x0 AB ω B |B|=l−1 j=1, j ∈B /
+
|B|=l
+ p − 1 2k βj βj (−1)j ∂rβj AB + AB ωB\{βj } Pk (x) = 0. r β j j=1
l+1 |B|=l+1
Proof. We first observe that
∂x(βj ) ω B = (−1)j−1 ∂x(βj ) ωβj ω B\{βj } = (−1)j
whence ∂x ω B =
l
(−1)j
j=1
Applying the Leibniz rule (2.2) yields ∂x AB ω B Pk (x)
(pβj − 1) ω B\{βj } , rβj (pβj − 1) ω B\{βj } . rβj
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d l (p − 1) β j = ∂rj AB ω j ω B + AB (−1)j ω B\{βj } Pk (x) r β j j=1 j=1 l
k β +(−1)l AB ωB ∂x Pk (x) + 2AB (−1)j j ω B\{βj } Pk (x) rβj j=1 d ∂rj AB ω j ω B = j=1, j ∈B /
+ p − 1 2k β β j j + (−1)j ∂rβj AB + AB ω B\{βj } Pk (x). r β j j=1 l
Hence ∂x
AB ω B
Pk (x)
B
d =
∂rj AB ω j ω B + ∂x0 AB ω B
j=1, j ∈B /
B
+ p − 1 2k β β j j + (−1)j ∂rβj AB + AB ω B\{βj } Pk (x) r β j j=1 d d = ∂rj AB ω j ω B + ∂x0 AB ωB l
l=0
|B|=l−1 j=1, j ∈B /
|B|=l
2kβj + pβj − 1 j + (−1) ∂rβj AB + AB ω B\{βj } Pk (x), rβj j=1 l+1
|B|=l+1
which proves the lemma. We can now prove Theorem 3.1.
Proof. Our proof starts with the observation that m − d is even if p1 , . . . , pd are d odd. Indeed, observe that m − d = s=1 (ps − 1). Since each ps − 1 is even, so is m − d. Next, by Lemma 3.3, we have k+ m−d 2 ∆x GB (x0 , r1 , . . . , rd ) ω B Pk (x) B
= (2k + m − d)!!
B
AB (x0 , r1 , . . . , rd ) ω B
Pk (x),
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with AB =
d
Drs
s=1 s∈B /
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l p s − 1 rβc pβ − 1 D ks + kβc + c {GB }. 2 2 c=1
Statements (i) and (ii) of Lemma 3.2 imply that for each l = 0, . . . , d d d d ps − 1 ∂rj AB ω j ωB = Drs ks + 2 s=1 j=1 j=1 |B|=l−1
|B|=l−1
j ∈B /
×
l−1
pβ − 1 pj − 1 kβc + c ∂rj Drj kj + {GB } ωj ω B 2 2 d ps − 1 Drs ks + 2 s=1
D rβc
c=1
=
/ j ∈B / s∈B∪{j}
d
|B|=l−1 j=1 / j ∈B / s∈B∪{j}
×
l−1
D
rβc
c=1
and
pβ c − 1 pj − 1 rj ∂rj GB ω j ω B kβc + kj + D 2 2
2kβj + pβj − 1 (−1) ∂rβj AB + AB ωB\{βj } rβj |B|=l+1 j=1
l+1 d ps − 1 j = (−1) Drs ks + 2 j=1 l+1
j
|B|=l+1
×
l+1
s=1, s∈B /
D rβc
c=1 c=j
pβ j − 1 pβ c − 1 rβj kβc + kβj + ∂rβj D {GB } 2 2 pβ j − 1 ωB\{βj } kβj + {GB } 2 d ps − 1 Drs ks + 2 s=1
(2kβj + pβj − 1) rβ + D j rβj =
l+1 |B|=l+1 j=1
×
l+1 c=1 c=j
D rβc
(−1)j
s∈B\{β / j}
pβ c − 1 kβc + {∂rβj GB } ωB\{βj } . 2
From the above it follows that for each l = 0, . . . , d d ∂rj AB ωj ω B + ∂x0 AB ωB |B|=l−1 j=1, j ∈B /
|B|=l
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2k + p − 1 β β j j (−1)j ∂rβj AB + AB ωB\{βj } Pk (x) r β j j=1
l+1
+
|B|=l+1
1 k+ m−d ∆x 2 = (2k + m − d)!!
l+1
+
|B|=l+1 j=1
d
∂rj GB ω j ωB +
|B|=l−1 j=1 j ∈B /
∂x0 GB ω B
|B|=l
(−1)j ∂rβj GB ωB\{βj } Pk (x)
= 0, where the last equality is a consequence of (3.1). The theorem immediately follows from Lemma 3.4. It is worth pointing out that the conclusion of Theorem 3.1 does not hold in general if some of the integers ps are even. For instance, for d = p1 = p2 = 2, k = 0 and G(y0 , y1 , y2 ) = y1 + y2 − y0 E1 − y0 E2 , we have 1 1 ∂x ∆x (r1 + r2 − x0 ω 1 − x0 ω 2 ) = x0 + = 0. r13 r23 We conclude the paper with some examples. Example. Let G(y0 , y1 , . . . , yd ) =
d
ys − y0
s=1
d d
ys Ej .
j=1 s=1 s=j
It is easy to check that d s=1
Drs
ps − 1 ks + 2
d
rs
= (−1)k+
m+d 2
s=1
d (2ks + ps − 4)!! s=1
rs2ks +ps −2
,
d ps − 1 pj − 1 rj Drs ks + rs kj + D 2 2 s=1 s=1 d
s=j
s=j
= (−1)k+
m+d 2 −1
d (2kj + pj − 2)!! (2ks + ps − 4)!! 2kj +pj −1
rj
s=1 s=j
rs2ks +ps −2
.
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Therefore
d d d 1 k+ m−d rs − x 0 rs ω j Pk (x) ∆x 2 C s=1 j=1 s=1 s=j
d = s=1
d
d
(j)
1 1 x P (x), + x0 (2kj + pj − 2) 2ks +ps −2 2kj +pj k rs2ks +ps −2 r rj s=1 s j=1
x(s) = 0,
s=j
where C = (−1)k+
m+d 2
d
(2k + m − d)!!
(2ks + ps − 4)!!
s=1
is a monogenic function. Example. Consider the fundamental solution G(y0 , y1 , . . . , yd ) =
y0 − (y02 +
d j=1
yj Ej
j=1
yj2 )
d
d+1 2
of ∂y0 + ∂y . As d
ps − 1 1 Drs ks + d d+1 2 (x20 + j=1 rj2 ) 2 s=1 1 (2k + m − 1)!! d m+1 , 2 (d − 1)!! (x0 + j=1 rj2 )k+ 2 d − 1 ps − 1 p r j j Drs ks + Drj kj + d+1 2 2 (x20 + d rj2 ) 2 s=1 = (−1)k+
m−d 2
j=1
s=j
= (−1)k+
m−d 2
(2k + m − 1)!! rj d m+1 , 2 (d − 1)!! (x0 + j=1 rj2 )k+ 2
we obtain the well-known monogenic function + * 1 k+ m−d x x ∆x 2 (x) = P Pk (x), k C |x|d+1 |x|2k+m+1 where C = (−1)k+
m−d 2
(2k + m − d)!!
(2k + m − 1)!! . (d − 1)!!
Example. Consider the steering monogenic function (see [11]) G(y0 , y1 , . . . , yd ) = d
E1
2 j=2 yj
d−3 2
x = 0,
d (d − 3) j=2 yj Ej (y1 + y0 E1 ) − . d−1 2 d 2 2 y j=2 j
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A direct computation shows that d ps − 1 p1 − 1 1 r1 Drs ks + k1 + D d d−3 2 2 ( j=2 rj2 ) 2 s=2 = (−1)k+
m−d 2
(2k1 + p1 − 2)!!
r12k1 +p1 −1
2(k − k1 ) + m − p1 − 4 !! 1 × d m−p1 −2 , 2 k−k (d − 5)!! ( j=2 rj ) 1 + 2 d ps − 1 pj − 1 r1 rj rj Drs ks + kj + D d d−1 2 2 ( rj2 ) 2 s=1 j=2
s=j
= (−1)k+
m−d 2 +1
(2k1 + p1 − 4)!!
r12k1 +p1 −2
2(k − k1 ) + m − p1 − 2 !! × d (d − 3)!! (
rj
2 k−k1 + j=2 rj )
and
d s=1 s=1,j
m−p1 2
,
ps − 1 p1 − 1 Drs ks + Dr1 k1 + 2 2 pj − 1 rj ×D kj + d d−1 2 ( j=2 rj2 ) 2 rj
= (−1)k+
m−d 2
(2k1 + p1 − 2)!!
r12k1 +p1 −1
2(k − k1 ) + m − p1 − 2 !! × d (d − 3)!! (
rj
2 k−k1 + j=2 rj )
We thus get the monogenic function m−d 1 k+ 2 ω1 ∆x d−3 2 C (|x| − |x(1) |2 ) 2
=
(d − 3) x − x(1) (r1 + x0 ω 1 ) − d−1 2 (|x|2 − |x(1) |2 ) 2 (2k1 + p1 − 2)x(1) r12k1 +p1
m−p1 2
.
Pk (x)
1 m−p1 −2
(|x|2 − |x(1) |2 )k−k1 + 2
2(k − k1 ) + m − p1 − 2 1 (2k1 + p1 − 2)x0 x(1) + − 2 r12k1 +p1 −2 r12k1 +p1
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×
(x − x(1) ) (|x|2 − |x(1) |2 )k−k1 +
m−p1 2
Pk (x),
where C = (−1)k+
m−d 2
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(2k + m − d)!!(2k1 + p1 − 4)!!
2(k − k1 ) + m − p1 − 4 !! . (d − 5)!!
References [1] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Research Notes in Mathematics, 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. [2] W. K. Clifford, Applications of Grassmann’s Extensive Algebra. Amer. J. Math. 1 no. 4 (1878), 350–358. [3] R. Delanghe, F. Sommen and V. Souˇcek, Clifford algebra and spinor-valued functions. Mathematics and its Applications, 53, Kluwer Academic Publishers Group, Dordrecht, 1992. [4] R. Fueter, Die funktionentheorie der differentialgleichungen ∆u = 0 und ∆∆u = 0 mit vier variablen. Comm. Math. Helv. 7 (1935), 307–330. [5] K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter’s theorem. Methods Appl. Anal. 9 no. 2 (2002), 273–289. [6] G. Laville and E. Lehman, Analytic Cliffordian functions. Ann. Acad. Sci. Fenn. Math. 29 no. 2 (2004), 251–268. [7] G. Laville and I. Ramadanoff, Holomorphic Cliffordian functions. Adv. Appl. Clifford Algebras 8 no. 2 (1998), 323–340. [8] D. Pe˜ na Pe˜ na, T. Qian and F. Sommen, An alternative proof of Fueter’s theorem. Complex Var. Elliptic Equ. 51 no. 8-11 (2006), 913–922. [9] D. Pe˜ na Pe˜ na and F. Sommen, A generalization of Fueter’s theorem. Results Math. 49 no. 3-4 (2006), 301–311. [10] D. Pe˜ na Pe˜ na and F. Sommen, Some power series expansions for monogenic functions. Comput. Methods Funct. Theory 7 no. 1 (2007), 265–275. [11] D. Pe˜ na Pe˜ na and F. Sommen, On Steering Monogenic Functions. Commun. Math. Anal. 4 no. 2 (2008), pp. 61–66. [12] D. Pe˜ na Pe˜ na and F. Sommen, Special power series expansions of monogenic functions around surfaces of codimension two. Math. Methods Appl. Sci. 32, No. 6 (2009), 631-639 [13] D. Pe˜ na Pe˜ na and F. Sommen, Monogenic Gaussian distribution in closed form and the Gaussian fundamental solution. To appear in Complex Variables and Elliptic Equations. [14] T. Qian, Generalization of Fueter’s result to Rn+1 . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8 no. 2 (1997), 111–117. [15] T. Qian, Fourier analysis on starlike Lipschitz surfaces. J. Funct. Anal. 183 no. 2 (2001), 370–412. [16] T. Qian and F. Sommen, Deriving harmonic functions in higher dimensional spaces. Z. Anal. Anwendungen 22 no. 2 (2003), 275–288.
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[17] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 23 (1957), 220–225. [18] F. Sommen, On a generalization of Fueter’s theorem. Z. Anal. Anwendungen 19 no. 4 (2000), 899–902. Dixan Pe˜ na Pe˜ na and Frank Sommen Department of Mathematical Analysis Ghent University Galglaan 2, 9000 Gent Belgium e-mail:
[email protected];
[email protected] [email protected] Received: November 5, 2008. Accepted: January 25, 2009.