Analytic functions and Milne Thomson Algorithm

A complex number looks like \(a+ib\), where \(a\) and \(b\) are real numbers and \(i^2=-1\). If we denote the set of complex number by \(\mathbb{C}\), then any function \(f:D \to \mathbb{C}\) would be referred to as a complex valued function on a subest \(D\) of complex numbers. Suppose \(D\) is such that taken any point \(a\), there is a circular disc of some radius centered at \(a\) lying entirely in \(D\), then \(a\) would be called an interior point of \(D\). If every point of \(D\) is interior, then \(D\) would be called an open set.

Now, a function \(f:D\to \mathbb{C}\), where \(D\subset \mathbb{C}\) is said to 'analytic' on \(D\) if \(f\) is differentiable at each point of \(D\) and the derivative is continuous. That is, at each point \(z_0\) of \(D\), \(\lim\limits_{z\to z_0}\dfrac{f(z)-f(z_0)}{z-z_0}\) exists and \(f^{\prime}(z)\) is continuous at \(z_0\).

Now, on a domain of the form \(\{z:|z|<\rho\}\) (i.e. a circular disc centered at origin with radius \(\rho\)), if the the real part \(u\) (or imaginary \(v\)) part of an analytic function \(f(z)\) is give, then we can find the imaginary part (or real part) of it as follows. The technique is due to Milne Thomson and hence is named after him as Milne Thomson algorithm.

1. First express \(f(z) = u(x,y)+iv(x,y)\) which implies \(f^{\prime}(z) = u_x+iv_x\) which from Cauchy-Riemann equations take the form \(f^{\prime}(z) = u_x-iu_y\) (or \(f^{\prime}(z) = v_y+iv_x\))
2. Put \(x=z\) and \(y=0\) in the expression \(f^{\prime}(z)=u_x-iu_y\) (or in the expression \(f^{\prime}(z) = v_y+iv_x\))
3. Integrate the expression the obtained after the replacements.
4. If we are finding the imaginary part, the constant of integration would be \(ic\), where \(c\) is a real number. If we are finidng the real part, then \(c\) is the constant of integration, where \(c\) is a real number