A generalization of a Cullen's Integral Theorem for the quaternions
aa r X i v : . [ m a t h . C V ] A p r A generalization of a Cullen’s Integral Theoremfor the quaternions
Daniel Alay´on-Solarz ([email protected])November 14, 2018
Abstract
We discuss the proof of a certain integral theorem obtained by C. G.Cullen, originally stated on the class of the analytic intrinsic functions onthe quaternions. It is shown that this integral theorem is true for a largerclass of quaternionic functions.
Let H be the algebra of the quaternions and let p be a quaternion, then p canbe written as p := t + xi + yj + zk. Let f be a quaternion-valued function of a single quaternionic variable. Considerthe class of complex-like quaternionic functions such that a member f can bewritten as f = u + ιv, where u and v are real functions and ι is defined as ι := xi + yj + zk p x + y + z . In particular the identity function, sending one quaternion onto itself is complex-like and thus the quaternion p can be written as: p = t + rι, where r := p x + y + z . Recall the left-Fueter operator is given by: D l := ∂∂t + i ∂∂x + j ∂∂y + k ∂∂z . For the rest of this paper, the class of complex-like quaternionic functions sat-isfying D l f = − vr Hyperholomorphic . Note that ι can be parametrizedby spherical coordinates ι = (cos α sin β, sin α sin β, cos β ) . The coordinate system based in the variables ( t, r, α, β ) is especially well suitedto study the interplay between the Fueter operator and the complex-like quater-nionic functions. The Fueter operator in this coordinate system has the form: D l = ∂∂t + ι ∂∂r − r ∂∂ l ι , where ∂∂ l ι := ( ι α ) − ∂∂α + ( ι β ) − ∂∂β . and ι α and ι α represent the derivatives of ι with respect to α and β respectively.Using this coordinate system the following characterization of Hyperholomor-phic functions can be proved: Proposition 1
A function f = u + ιv is hyperholomorphic if and only if u and v satisfy: ∂u∂t − ∂v∂r = 0 ,∂v∂t + ∂u∂r = 0 , and ∂v∂α (sin β ) − + ∂u∂β = 0 ,∂u∂α (sin β ) − − ∂v∂β = 0 . It is well known that regular functions in the Fueter sense, that is, quater-nionic null-solutions to the Fueter operator, are in general not closed under thequaternionic product. However non-zero hyperholomorphic functions form amultiplicative group:
Proposition 2
The sum and product of two hyperholomorphic functions is hy-perholomorphic. If a function is hyperholomorphic and non-zero then its alge-braic inverse is hyperholomorphic.
A important property of hyperholomorphic functions is that they generalize theFueter’s Theorem [3]:
Proposition 3
Let f be a hyperholomorphic function. Then D l ∆ f = D r ∆ f = 0 . Where by D r we denote the right-Fueter operator. Fueter’s theorem was origi-nally stated for the smaller class of quaternionic functions obtained by rotatingaround the real axis a complex analytic function.The purpose of this paper is to show how hyperholomorphic functions satisfythe following integral theorem given by Cullen in [1].2 roposition 4 Let f = u + ιv be an hyperholomorphic function and let K anysmooth, simple closed hypersurface in H the quaternionic space, disjoint fromthe real axis, K ∗ being the interior of K . Let n ( p ) = n + n i + n j + n k where ( n , n , n , n ) is the unit outer normal to K at p . Then, Z K n ( p ) f ( p ) 1 r dS K = − Z K ∗ u ιr dV. where dS K is the element of surface area on K . This proof we will consider is the same Cullen gave in [1] and it requires twopreliminar lemmas. Our contribution consists on the observation that one ofthese lemmas used by Cullen and satisfied by the class of analytic intrinsic func-tions is actually a definition of the larger class of hyperholomorphic functions.
Proposition 5 (Cullen’s lemma) A function f = u + ιv a hyperholomorphic ifand only if: D l ( fr ) = − r uι. Proof.
We first assume f is hyperholomorphic, then: D l ( fr ) = ( ∂∂t + ι ∂∂r − r ∂∂ l ι )( u + ιvr )= ( ∂∂t + ι ∂∂r )( fr ) − r v = 1 r ( ∂f∂t + ι ∂f∂r ) − r ( uι − v ) − r v = − r uι. Note how we use the fact that since f is hyperholomorphic then: ∂f∂ l ι = 2 v which in turn is a consequence of the hyperholomorphic functions being Cullen-regular [2], that is it satisfies the following equation:( ∂∂t + ι ∂∂r ) f = 0Note that for all functions f = u + ιv : D l ( fr ) = 1 r ( ∂∂t + ι ∂∂r − r ∂∂ l ι ) f − r ( ιf )so in particular if f satisfies the Cullen lemma then:1 r ( ∂∂t + ι ∂∂r − r ∂∂ l ι ) f − r ( ιf ) = − r uι implies 1 r D l f = − r v which for r = 0 is the condition for hyperholomorphicity. We continue with thesecond lemma: 3 roposition 6 Let ( n , n , n , n ) be the outward unit normal of K , then Z K ∗ ( ∂f ∂t + ∂f ∂x + ∂f ∂y + ∂f ∂z ) dV = Z K ( f n + f n + f n + f n ) dS k where f i are differentiable quaternionic functions. Proof.
Following Cullen, this is an application of the Gauss Theorem in fourdimensions for the components of f i .We are now ready to prove our main result, for this it suffices to show that for f = u + ιv an hyperholomorphic function we have: Z K n ( p ) f ( p ) 1 r dS K = Z K ( n fr + n ifr + n jfr + n kfr ) dS K = Z K ∗ D l ( fr ) dV = − Z K ∗ u ιr dV. And it is proved.If one starts with the class of right-hyperholomorphic functions, that is thosefunctions of the form f = u + ιv that satisfy D r f = − vr then the integral theorem would read: Z K f ( p ) n ( p ) 1 r dS K = − Z K ∗ u ιr dV. The class of functions that are both left- and right-hyperholomorphic is notempty. For example the class of analytic instrinsic functions on the quaternionsis of this form.