实部和虚部：复变函数的实变量实现Real and imaginary parts
文章探讨了如何通过实变量的函数组合来表示复变函数，即找到实函数 u(x, y) 和 v(x, y)，使得 f(x + iy) = u(x, y) + i v(x, y)。作者引用了 Henry Baker 的方法，展示了如何将复数运算分解为实数运算，并提供了具体实现思路。这种方法简化了复变函数的计算过程，尤其适用于数值分析和工程应用。核心观点是：复变函数问题可以通过实变量函数高效求解。
John
The previous post announced some notes I wrote up based on an article by Henry Baker implementing functions of a complex variable in terms of functions of a real variable. That is, it finds functions u(x, y) and v(x, y) such that
f(x + iy) = u(x, y) + i v(x, y)
where x, y, u, and v are all real-valued. Not only that, but if f is an elementary function, so are u and v. Here “elementary” has a technical meaning, but essentially it means functions that you could evaluate on a scientific calculator. A couple of the functions might be unfamiliar, such as sgn and atan2, but there are no functions like the gamma function that are defined in terms of integrals.
One application of Baker’s equations would be to bootstrap a math library that doesn’t support complex numbers into one that does. But the equations could be useful in pure math when you’d like to have a convenient expression for the real or imaginary part of a function.
The real and imaginary parts of a complex analytic function are harmonic functions. So the functions on the right hand side of Baker’s equations satisfy Laplace’s equation:
uxx + uyy = 0
and
vxx + vyy = 0.
Furthermore, the functions u and v form harmonic conjugate pairs, meaning each is the Hilbert transform of the other.
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