Talk:Hypercomplex analysis

Biquaternion thoughts

The following was moved from Biquaternion. It may contribute to hypercomplex analysis as a sample work:

Biquaternion functions

Functions of Complex Quaternions

The exponential function is well-defined by its power series, which converges over the entire domain, even for complex quaternions. Since the basic circular and hyperbolic functions cos, sin, cosh, sinh are linear combinations of exponential functions, they too are well-defined. Some preliminaries: In this section I J K are used for the basis quaternions so that i may be used as the square root of -1. Here I I = -1, I J = K, etc. Let x = a + b I + c J+ d K be a complex quaternion with complex coefficients a, b, c, d and with norm a² + b² + c² + d². The norm of a product is the product of the norms. The complex quaternions do not form a normed division algebra since there are non-zero elements with zero norm. These elements have no inverse. A simple expression for exp(x) is sought. The vector part v = b I + c J+ d K has the property that its square is the scalar -(b² + c² + d²), which is the negative of its norm. The vector quaternion v is called basis-like if its square is -1. Let the complex s be the square root of - v v. Any power series in the vector part v can be expressed as a sum of an even power series in s and an odd power series in s which multiplies the vector part v divided by s. If these two power series are those of common functions, then this gives a simple way to evaluate. Now exp(a + b I + c J + d K) = exp(a) exp(b I + c J + d K) since a is in the complex sub-algebra which commutes with the complex quaternion algebra. The second exponential factor can be expressed as a cos(s) term plus a sin(s)/s term multiplying the vector part v. This assumes s to be non-zero. For the case for which the vector part N of the quaternion has zero norm, exp(a + b N) = exp(a) ( 1 + b N).

The sqrt function can be handled even though it has a branch cut. There are more than two multiple values. For the complex numbers, any multiple by a vector quaternion with square +1 such as i I is also a possible value. The case for which the vector part has square zero is a special case. Indeed, a quaternion which is a non-zero vector quaternion with square zero has no square root. For a quaternion with a vector part having non-zero norm, such as a + b I with b non-zero, there are four possibilities for the multiplying factor, namely 1, -1, i I, and -i I. Consider the complex quaternion x = a + b I. The vector part, assumed to have a non-zero norm, can always be scaled so that it is basis-like with norm +1 with the complex scaling factor being absorbed into b. No generality is lost by representing this quaternion vector by I since only the property that its square is -1 is needed in this discussion. Its square root must have the form c + d I. This gives two complex equations for the two complex unknowns c and d. If a is non-zero, its square root can be factored out so that now the square root to be found is of 1 + A I. Within a common sign, c = (sqrt(1+ i A)+sqrt(1- iA))/2 and d= -i (sqrt(1+iA)-sqrt(1-iA))/2. Notice that it doesn’t matter which sign is picked for sqrt(1+iA) and which sign is picked for sqrt(1-iA) so long as the choices are consistent. If a is zero, then the square root of I needs to be found. It is (1+I)/sqrt(2 ) within a sign. As an example, sqrt(2 + 3*i*I + J) = 1.09868 + i * 0.45509 + (0.482696 + i * 1.16533) * I + (0.388443 + i * -0.160899) * J. The special case a + b N where a is non-zero and N is a null vector quaternion has a square root c + d N given by c = sqrt(a) and d = b / (2* sqrt(a)).

The logarithm function log(1+x) is not well defined and its power series only converges in a region near the origin. As a cautionary example, consider log(-1). Some possible values are pi*i, pi*I, pi*J, pi*K, and pi times any quaternion vector with norm +1. Admittedly this is a pathological case. If x has the form c + d I, then exp(x) always has the form a + b I. It is assumed that the vector part has a non-zero norm so that it can be scaled to be basis-like with norm +1 and be represented by I. It is only for complex numbers with b equal to zero that the pathology arises. Otherwise only the commutative subalgebra generated by 1, i, and I need be considered. For instance, log(I) has the values (pi/2)*I + 2 m pi i + 2 n pi I, where m and n are integers. There will always be a grid of solutions 2 m pi i + 2 n pi I centered on a particular solution. Let b/a = tan(theta) where theta is complex. Now exp(theta I) = cos(theta) + sin(theta) I even for complex theta. So 1 + (b/a) I = cos(theta) ( 1 + tan(theta) I). So c = log( a / cos(theta) ) and d = atan(b/a). As an example, log(2+(1+i)*I) = 0.748933 + i * 0.231824 + (0.553574 + i * 0.402359) * I. The log of a quaternion that is a null vector does not exist. For N a null vector quaternion, log(1 + N) = N.

The circular and hyperbolic functions are defined in terms of the exp function. Examination of the cosine and sine functions and their inverses demonstrates the important concepts for all. They are defined by cos(x) = (exp(i x) + exp(-i x))/2 and sin(x) = (exp(i x) - exp(-i x))/(2 i). Using DeMoivre's formula exp(i x) = cos(x) + i sin(x) and the identity cos²(x) + sin²(x) = 1 lets x be expressed in terms of either cos(x) or sin(x) using the sqrt and log functions, giving their respective inverses arccosine and arcsine. Finding those values of x for which cos(x) or sin(x) have the same values reveals the multi-valued behavior of their inverses. Except for special cases, x = a + b I, where I is a basis-like vector quatornion of norm +1 and b is non-zero. For integers m and n, adding m pi + n pi i I to x takes exp(i x) into itself for even m + n and into its negative for odd m + n. The even case leaves both cos(x) and sin(x) unchanged. The odd case together with negating x leaves sin(x) unchanged. A special case is when x = a + N, where N is a non-zero vector quaternion having zero norm. The 2D grid then becomes the 1D grid m pi. The other special case is when x = a, where a is complex. Then adding to x any term of the form n pi i I, where I is any basis-like vector quaternion of norm +1, takes exp(i x) into itself for even n and into its negative for odd n. The hyperbolic functions are just the circular functions with the complex plane rotated by 90 degrees.

The complex quaternion function exp can be used to perform Lorentz boosts. This cannot be done with the real quaternions. The following material adds briefly to the sections Relation to Lorentz transformations and As a composition algebra in this article. A 4-vector in special relativity has the form X = t + x i I + y i J + z i K, where t, x, y, and z are real. Its norm is the Minkowski invariant t² - x² - y² - z². Let a ^T superscript denote a "transpose" operation taking I into -I, J into -J, and K into -K. The reason for the "transpose" name is that I, J, and K can be represented as 4x4 real anti-symmetric matrices and that is what the matrix transpose does. The transpose of a product is the product of the transposes in reverse order, as in matrix algebra. Let q* denote complex conjugation of the quaternion q. This operation does not do anything to I, J, and K. This definition is again reasonable since I, J, K can be expressed as 4x4 anti-symmetric real matrices. Let an overbar denote the complex conjugate transpose. If q = a + b I + c J + d K, then q = a* - b* I - c* J - d* K. Note that X = X. Let q have norm 1. Consider X' = q X q. If q has norm 1, then since the complex quaternions are a composition algebra, X' and X have the same norm. Also X' has the same form as X with real scalar part and imaginary spatial part. The complex quaternion B = exp(i I α/2) = cosh(α/2) + i sinh(α/2) I does a boost in the x direction using X' = B X B. Here tanh(α) = v/c is the boost. Similarly, the quaternion R = exp(I θ/2) = cos(θ/2) + sin(θ/2) I does a spatial rotation about the x axis by angle θ using X' = R X R. R is also a real quaternion, unlike the boost quaternion. An aside on the 4x4 real anti-symmetric matrix representation: In terms of the Pauli spin matrices, I, J, and K multiply as i σ_x, -i σ_y, and i σ_z, respectively. The unit imaginary i can be represented as a 2x2 real antisymmetric matrix, zero can be represented as the 2x2 zero matrix, and 1 can be represented as the 2x2 identity matrix. This gives the 4x4 matrix representation. Any group of 2x2 complex matrices can be made 4x4 real with the same group multiplication properties. The group here are the matrices i σ_x, -i σ_y, and i σ_z, the 2x2 identity matrix, and their negatives. One question remains: Suppose q has norm one. What Lorentz transformation is it associated with? It always represents some Lorentz transformation since X' = q X q is a linear transformation of X, preserves the Minkowski invariant, and preserves the form X = X. Any proper Lorentz transformation can be represented uniquely either as a boost B₁ followed by a rotation R₂ or as a rotation R₁ followed by a boost B₂. Note that B = B and that R R = 1. To be definite, let q = B R. Then q q = B R R B = B B = B B. Its square root gives B. Then R = B⁻¹ q. As a check, let Q = q q. Note that Q = Q and that Q has a positive real scalar part and a pure imaginary spatial part. Its norm is one since q has norm one. So its scalar part is not only positive real but is also greater than or equal to one. So it has the form of a boost. As a further check, R R = q B⁻¹ B⁻¹ q = q (q q)⁻¹ q = 1. That R R = 1 means that R is a rotation except for a possible phase factor. That B R has norm 1 means that the phase factor is 0 or 180 degrees. That doesn't matter. Changing the sign of q still gives the same Lorentz transformation.

References

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