The g-analytic Function Theory and Wave Equation

In this paper we develop a g-analytic function and a g-harmonic function theory for one-dimensional wave equation in the Minkowski space. In terms of the Minkowskian polar coordinates we can derive a set of complete hyperbolic type Trefftz bases, which can be transformed to polynomials as the bases for a trial solution of wave equation. The Cauchy-Riemann equations and the Cauchy theorem for g-analytic functions are proved, and meanwhile the existence of Cauchy integral formula is disproved from the non-uniqueness of the Dirichlet problem for wave equation under the boundary conditions on whole boundary, which is also known as the backward wave problem (BWP). Examples are used to demonstrate these results.


Introduction
The one-dimensional wave equation is given by where c is the speed of wave propagation, and the domain Ω may be bounded or unbounded.The wave propagation problems have attracted a lot of attentions since early last century and have been studied theoretically, computationally and experimentally due to its vital role in physical and engineering applications (Bleistein, 1984).
It is known that Equation (1) has a general solution: where f and g are twice differentiable functions.It is easily verified that the necessary and sufficient conditions for u to be the form (2) it must satisfy the diamond rule: for any characteristic rectangle ABCD with A at the top corner, B at the left corner, C at the bottom corner, and D at the right corner.However, for the Dirichlet problem of wave equation, for example, it is a classical ill-posed problem, which has been studied by Bourgin & Duffin (1939); John (1941); Fox & Pucci (1958); Dunninger & Zachmanoglou (1967); Abdul-Latif & Diaz (1971); Papi Frosali (1979); Levine & Vessella (1985); Vakhania (1994); Kabanikhin & Bektemesov (2012).They asserted that when α is a rational number the solution is not unique.

Let
u 1 = ∂u ∂t , u 2 = ∂u ∂x ; we also come to a coupled hyperbolic system of first-order partial differential equations: of which Sobolev (1956) has studied the ill-posed property under imposed boundary conditions.
For wave equation (1), we will verify that u has a counterpart v, and they satisfy the newly defined Cauchy-Riemann equations as the pair in Equation ( 4).The Dirichlet problem of wave equation will be discussed from a different viewpoint in this paper.
The remaining portion of this paper is arranged as follows.For the purpose of comparison we introduce the Laplace equation and the analytic function theory of complex function in Section 2, where we propose some new problems for wave equation.In Section 3 we introduce a new g-number, corresponding to the complex number, in the Minkowksi space as a frame to study the wave equation.The Minkowksian polar coordinates of wave equation are developed, and the new Trefftz bases are derived.In Section 4 we develop a new g-analytic function theory for the g-function, and the Cauchy-Riemann equations are introduced for wave equation.The Cauchy theorem is derived in Section 5 for the g-analytic function.The existence of Cauchy integral formula is disproved by using the result from the Dirichlet problem of wave equation.Finally, some conclusions and the analogies between the Laplace equation and the wave equation are addressed in Section 7.
Let z = x + iy be a complex number, and f (z) = u(x, y) + iv(x, y) be an analytic function in a complex domain C. It is well known that the Cauchy-Riemann equations hold (Marsden, 1973): They imply both u(x, y) and v(x, y) satisfying the Laplace equation ( 5).When u is usually called the harmonic function in complex theory, v is called the conjugate harmonic function.
In the present paper we propose the problems: Are that there exist an analogous basis to Equation ( 6) and the Cauchy-Riemann pair (8) for wave equation ( 1)?In order to reply these problems we need to recast wave equation (1) into the one in the Minkowskian space-time domain.

Mathematical Preliminaries of g-Integral
Let us mention the Minkowski space M 1,1 and the rotation in that space.Liu (2000) has introduced the g number w = x + gy, where 1 and g are bases of a Jordan algebraic system which obey the following binary product rule: Using g 2 := g • g = 1 in the Taylor series expansion of e gθ , which corresponds to the famous formula: where i 2 = −1 and θ ∈ R.
For the complex number z = x + iy we can view it as a point in the Euclidean space R 2 , and z ′ = x ′ + iy ′ = e iθ (x + iy) can be viewed as a rotation in R 2 by using In contrast, for the g number w = x + gy we can view it as a point in the Minkowski space M 1,1 , and w ′ = x ′ + gy ′ = e gθ (x + gy) can be deemed as a rotation in M 1,1 : As pointed out by Liu (2002) the g-number bears certain similarity with the complex number, and it forms a Jordan algebra, which is a special case of the double numbers introduced by Yaglom (1968).The applications of the Jordan algebra can refer (Iordanescu, 2007;Iordanescu, 2009).
In the complex theory, for z, z 1 , z 2 ∈ C, we have where z = x − iy is the conjugate of z = x + iy and p is an integer.
Like the complex number, w = x − gy is the conjugate g-number of w = x + gy.Similarly, for w, w 1 , w 2 ∈ M 1,1 we can prove that by means of Quite different from the Euclidean length zz = x 2 +y 2 ≥ 0 for a point (x, y) in the Euclidean plane, the Minkowskian length ww = x 2 −y 2 may be positive, zero or negative.The formulae ( 17)-(20) may bear certain similarities with the formulae (13)-( 16).Besides that we will explore the similarity between the Laplace equation and wave equation (1).
As the definition for the complex function, a function with w = x + gy as an independent variable is called the g-function.
For the later use we may need to execute the integral of f (w) in the plane (x, y), which is called the g-integral.We use the following examples to demonstrate the integrals of g-function.
Example 1. Calculate the following complex integral along a unit circle: e −iθ e iθ idθ = 2πi.Vol. 7, No. 3;2015 It can be seen that θ plays both the roles as the integral variable and also the variable to represent the complex number in the polar coordinates.In the Euclidean space they are the same variable θ.
However, for the g-integral along a unit circle we cannot use the Minkowskian polar coordinates, because θ is a geometric variable in the Euclidean space to present the angle, not the Minkowskian polar angle variable as that used in the g-number.By using x = cos θ and y = sin θ, which must be viewed as integral parameters for the g-integral, we can do The above two integrals are similar.However, the following two integrals are totally different.
Example 2. Along a unit circle we have due to zz = 1.For the corresponding g-integral of the same function we have + tan( 2θ) This example shows that the integrals of complex function and g-function have quite different integral behavior.
In Section 5 we will demonstrate that the different behavior comes from the theory of singular point and analytic function.

The Wave Equation in the Minkowski
Through a suitable transformation of the t coordinate in Equation ( 1) it is always that (ct) 2 > x 2 for all (x, t) ∈ Ω with |x| bounded.For wave equation we prefer to employ the future cone in the time-like space as the problem domain, because time is always towards future with (ct) 2 > x 2 . Let x := r sinh θ, y := r cosh θ.
We must emphasize that y 2 − x 2 > 0 in the domain Ω, which meaning that (x, y) is a time-like vector in M 1,1 , and then the definitions ( 24) and ( 25) make sense.So a time-like point can be expressed as The pair (r, θ) may be named the Minkowskian polar coordinates, which are totally different from that in Equation (7).
According to Equation ( 23), the wave equation ( 1) can be written as By using Equations ( 24) and ( 25) we have ∂r ∂x = − sinh θ, ∂r ∂y = cosh θ, (28) Then through some operations we can derive Inserting them into Equation ( 27) and using cosh 2 θ − sinh 2 θ = 1 we arrive to It is interesting to note that the above equation bears certain similarity with the second equation in Equation ( 5).However, the third terms in these two equations are different with a minus sign.

The Trefftz Bases
Similar to that r k cos(kθ) and r k sin(kθ) satisfy Equation ( 5), we can prove that r k cosh(kθ) and r k sinh(kθ) identically satisfy Equation (32).We have Inserting them into Equation ( 32), ends the proof that r k cosh(kθ) is a solution of Equation ( 32).Similarly, we have Inserting them into Equation (32), we prove that r k sinh(kθ) is a solution of Equation ( 32).
So we have the following set of hyperbolic-type bases for wave equation (32): {1, r cosh θ, r sinh θ, . . ., r k cosh(kθ), r k sinh(kθ), . ..}, and we may call the method that employs the above bases to expand the trial solution of u the Trefftz method for wave equation.

The g-Analytic Function and Cauchy-Riemann Equations
The main results are given in the following theorems.For simple notation we let M + be the future cone in M 1,1 with (x, y) ∈ M 1,1 , y 2 − x 2 > 0 and y > 0. The results also hold for the space-like vector (x, y) ∈ M 1,1 , x 2 − y 2 > 0.
Theorem 1.Let f := u + gv : D ⊂ M + → M + be a given g-function, with D an open set in the future cone.Then f ′ (w 0 ) exists if and only if f is a differentiable function in the sense of real variable and, at (x 0 , y 0 ) = w 0 , u and v satisfy the Cauchy-Riemann equations: Thus, if ∂u/∂x, ∂u/∂y, ∂v/∂x and ∂v/∂y exist, then f is g-analytic on D. If f ′ (w 0 ) does exist, then Proof.For a separate and more direct proof is given here to show that if f ′ (w 0 ) exists, then u and v satisfy the Cauchy-Riemann equations (34).In the limit w − w 0 , let us take the special case that w = x + gy 0 .Then As x → x 0 we obtain the limit f ′ (w 0 ) = ∂u/∂x + g∂v/∂x.
On the other hand, let w = x 0 + gy.Then As y → y 0 we obtain the limit 1 g where g 2 = 1 was used.Thus, since f ′ (w 0 ) exists and has the same value regardless of how w approaches w 0 , we can obtain g ∂u ∂y By equating the real and g parts of this equation, we can derive the Cauchy-Riemann equations (34), and two formulas for f ′ (w) in Equation ( 35). 2 The analytic function is well-developed in the complex theory.In order to distinct the present analytic function with that in the complex theory, we call it the g-analytic function with its real and g parts satisfying the Cauchy-Riemann equations (34).Next we prove that Theorem 2. Let f : D ⊂ M + → M + be a g-analytic function, with f (w) = u(x, y) + gv(x, y).Then u and v satisfy wave equation ( 27): Proof.From Equation (34) it follows that Subtracting the above two equations we obtain Equation (36).On the other hand, Equation (34) renders A same procedure leads to Equation (37). 2 In the complex analytic function theory u and v are usually called harmonic function and conjugate harmonic function.By the same token, u and v in the g-analytic function theory might be called g-harmonic function and conjugate g-harmonic function.
For analytic function and g-analytic function, the Cauchy-Riemann equations in terms of (Euclidean and Minkowskian) polar coordinates are, respectively, From Equation ( 38), the second equation in Equation ( 5) follows.Similarly, from Equation (39), Equation (32) follows straightforward.
The most simple analytic function and g-analytic function are polynomial functions f (z) = z k and f (w) = w k .
Let z = x + iy = re iθ , and it is easy to derive Let w = y + gx = re gθ .From Equation ( 24) it is easy to derive We can express the Trefftz bases (33) for wave equation directly in terms of the above polynomials.However, the Trefftz bases (6) for the Laplace equation cannot be directly expressed in terms of polynomials; they must take the real and imaginary parts of z k .
We can decompose the wave equation ( 27) into By analogy to the Laplace equation we can derive a neater form: where w = y + gx and w = y − gx.Hence, the general solution of Equation ( 44) is u = ϕ(w) + ψ(w); that is, u = ϕ(y + gx) + ψ(y − gx).Since ψ is arbitrary we can take ψ(w) = ϕ(w).Then ψ(w) = ϕ(w) = ϕ(w), and u = 2Re[ϕ(w)].If we take ϕ as a g-analytic function and replace it by ϕ/2, we see that the real part of any ganalytic function is a g-harmonic function of x and y.The converse is also true: any g-harmonic function is the real part of a g-analytic function.

The Cauchy Theorem
The Cauchy theorem in complex theory is closely related to the exact function on the plane (x, y).Again we let M + be the future cone in M 1,1 with (x, y) ∈ M 1,1 , y 2 − x 2 > 0 and y > 0.
Theorem 3. Suppose that f (w) is a g-analytic function, with f ′ continuous, on and inside/outside the simple closed curve Γ ∈ M + .Then Due to the first Cauchy-Riemann equation ( 34) with u x = v y there exists a potential function dϕ(x, y) = ϕ x dx + ϕ y dy = vdx + udy; similarly, another potential function ψ exists, such that dψ(x, y) = ψ x dx + ψ y dy = udx + vdy due to u y = v x .The subscripts above denote the partial differentials.Then we have Also we can prove it by using the Green's Theorem to each integral: Both terms are zero by using the Cauchy-Riemann equations (34). 2 Example 2 (continued).Now it is a good position to demonstrate that the contour integral in Equation ( 21) for 1 z is non-zero, but the contour integral in Equation ( 22) for 1 w is zero.It is that 1 is not analytic at the singular point z = 0 which is inside the domain.Although the above u and v satisfy the Cauchy-Riemann equations ( 8): On the other hand, is a g-analytic function, due to y 2 − x 2 > 0 (or x 2 − y 2 > 0) and The domain is an exterior one which is outside the unit circle, such that the singular point w = 0 is not in the domain.
Corollary 1. Suppose that u is a g-harmonic function, on and inside/outside the simple closed curve Γ ∈ M + .Then where the g part is zero due to Γ u x dx + u y dy = Γ du = 0. Then we obtain Equation ( 50). 2 Can we find a g-analytic function f = u+gv, such that its real part u is a given g-harmonic function?Indeed we have Theorem 4. In a simply connected region D ∈ M + , if we have a given g-harmonic function u then we can find a g-analytic function f = u + gv at most differencing by a constant g part.
Proof.We have by assuming a g-analytic function f = u + gv.Then, where (x 0 , y 0 ) is a fixed point in D and c 0 is a constant.We need to check that v is the g-part of f .That is, we need to check that u and v satisfy the Cauchy-Riemann equations (34).By Equation ( 51) it is trivial.
We give an example to demonstrate Theorem 4.
Example 3. Let u = x 2 + y 2 + xy be a g-harmonic function, which satisfies wave equation ( 27).Then we take f = u + gv with Hence, this g-analytic function f = u + gv is given by As a continuation we give two examples to demonstrate the Cauchy Theorem 3.
Example 4. First we take as that given in the above with c 0 = 0.The closed curve is given by Γ ) Although the integral domain D passes across the boundary x = y of future cone, we still obtain the correct value of line integral.Indeed, the first two lines are in the space-like region, while the last two lines fall into the time-like region.At the crossed point it gives no contribution to the integral, and both in the space-like region and time-like region the Cauchy theorem holds.Thus we can say that the Cauchy theorem is applicable in the whole space M 1,1 .
Example 5. Then we take and Γ is the same that in Example 4. It is easy to check that the above f is a g-analytic function.We have 6.The Non-existence of Cauchy Integral Formula and Backward Wave Problem In complex theory the Cauchy integral is given by where f (ζ) is an analytic function, and z is a domain point inside a multiply-connected domain with boundary Γ.
For the g-analytic function f (w) we do not have a similar result.First we prove that Corollary 2. For D ∈ M + , w ∈ D being a g-number, consider a box with a center w 0 = y 0 + gx 0 and each side having a length 2ℓ such that the closed curve along the box is given by Γ Proof.We have This ends the proof. 2 The above situation is drastically different from that appears in the complex theory as given below.
Corollary 3. In a simply connected region D ∈ C, z ∈ D being a complex number, consider a box with a center z 0 = x 0 +iy 0 and each side having a length 2ℓ such that the closed curve along the box is given by Γ Proof.We have This ends the proof of Equation ( 54).For a clockwise integral we can obtain Equation (55) directly.2 Equation ( 55) is a key point to prove the Cauchy integral formula (52) in a multiply-connected domain for the complex analytic function.Similarly, Equation ( 53) is a key point to prove the following result in a multiplyconnected domain for the g-analytic function.
Theorem 5. Let D ∈ M + be a multiply-connected region with an outer/inner contour Γ 0 counter-clockwise and some inner/outer contours Γ i , i = 1, . . ., n clockwise.If f (w) is continuous in D and g-analytic in D, then for every point w 0 = y 0 + gx 0 ∈ D we have where which is g-analytic in D besides at the point w 0 .Let w 0 be a center of the box as constructed in Corollary 2, and meanwhile we let Γ ℓ be a box clockwise, which is inside D if ℓ is small enough.We apply Theorem 3 to the region enclosed by Γ and Γ ℓ , such that As shown in Corollary 2 the right-hand side is independent to ℓ, and when ℓ → 0 we have In view of Corollary 2 this theorem is proven.2 The situation of Equation ( 56) is drastically different from the Cauchy integral formula (52) in the complex theory.
To demonstrate this theorem we give the following example.
Example 6.Consider a unit box [0, 1] 2 with a contour , and let f (w) = w = y + gx be a g-analytic function.For any point w 0 outside the box we can evaluate We further demonstrate the different situations of the Cauchy integral formulae in the complex theory and in the g-number theory.Let us consider the one-dimensional wave equation: u(0, t) = u 0 (t), u(ℓ, t) = u ℓ (t), 0 ≤ t ≤ t f , (60)

Conclusions
We have developed a new theory of g-analytic function in the Minkowski space M 1,1 , whose foundation is the Jordan algebra.The real and g parts of the g-analytic function are respectively the g-harmonic function and conjugate g-harmonic function, satisfying the Cauchy-Riemann equations in the space M 1,1 and both being the solutions of wave equation.In summary, the analogies between the Laplace equation and wave equation are summarized as follows: We proved that the non-existence of the Cauchy integral formula in the space M 1,1 is closely related to the nonuniqueness of the solution of the Dirichlet problem for wave equation, which is an interior problem.However, the Cauchy integral formula in the space R 2 guarantees the unique solution of the interior problem for the Laplace equation.In the present theory the direct wave problem is therefore equivalent to finding in the specified domain a g-analytic function whose real part takes the given initial values and boundary values.

Figure 1 .
Figure 1.The recovery of a bell type initial velocity by using the Trefftz method to solve the Dirichlet problem of wave equation.