复距离势理论与双曲型方程
Complex-Distance Potential Theory and Hyperbolic Equations
摘要：
通过将欧氏距离函数全纯延拓到C^n，得到了R^n中位势理论的一个推广。由此产生的牛顿势由C^n中的扩展源分布D（z）产生，其对R^n的限制是δ函数。这为物理学中的扩展粒子提供了一个自然模型。在C^n中，被解释为复时空，D（z）充当传播子，从初始值生成波动方程的解。这给出了一个新的椭圆和双曲方程之间的连接，不承担解析的柯西数据。推广到Clifford分析，它诱导椭圆和双曲狄拉克方程的解之间的类似的连接。对时间相关的非齐次狄拉克方程组和麦克斯韦方程组有一个自然的应用，前面介绍的“电磁小波”就是一个例子。
英文摘要：
An extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n is the delta function. This provides a natural model for extended particles in physics. In C^n, interpreted as complex spacetime, D(z) acts as a propagator generating solutions of the wave equation from their initial values. This gives a new connection between elliptic and hyperbolic equations that does not assume analyticity of the Cauchy data. Generalized to Clifford analysis, it induces a similar connection between solutions of elliptic and hyperbolic Dirac equations. There is a natural application to the time-dependent, inhomogeneous Dirac and Maxwell equations, and the `electromagnetic wavelets' introduced previously are an example.
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