Milne-Thomson method for finding a holomorphic function

In mathematics, the Milne-Thomson method is a method for finding a holomorphic function whose real or imaginary part is given.^[1] It is named after Louis Melville Milne-Thomson.

Introduction

Let $z = x + i y$ and $\bar{z} = x - i y$ where $x$ and $y$ are real.

Let $f (z) = u (x, y) + i v (x, y)$ be any holomorphic function.

Example 1: $z^{4} = (x^{4} - 6 x^{2} y^{2} + y^{4}) + i (4 x^{3} y - 4 x y^{3})$

Example 2: $\exp (i z) = \cos (x) \exp (- y) + i \sin (x) \exp (- y)$

In his article,^[1] Milne-Thomson considers the problem of finding $f (z)$ when 1. $u (x, y)$ and $v (x, y)$ are given, 2. $u (x, y)$ is given and $f (z)$ is real on the real axis, 3. only $u (x, y)$ is given, 4. only $v (x, y)$ is given. He is really interested in problems 3 and 4, but the answers to the easier problems 1 and 2 are needed for proving the answers to problems 3 and 4.

Given both real and imaginary parts

Problem: $u (x, y)$ and $v (x, y)$ are known; what is $f (z)$ ?

Answer: $f (z) = u (z, 0) + i v (z, 0)$

In words: the holomorphic function $f (z)$ can be obtained by putting $x = z$ and $y = 0$ in $u (x, y) + i v (x, y)$ .

Example 1: with $u (x, y) = x^{4} - 6 x^{2} y^{2} + y^{4}$ and $v (x, y) = 4 x^{3} y - 4 x y^{3}$ we obtain $f (z) = z^{4}$ .

Example 2: with $u (x, y) = \cos (x) \exp (- y)$ and $v (x, y) = \sin (x) \exp (- y)$ we obtain $f (z) = \cos (z) + i \sin (z) = \exp (i z)$ .

Proof:

From the first pair of definitions $x = \frac{z + \bar{z}}{2}$ and $y = \frac{z - \bar{z}}{2 i}$ .

Therefore $f (z) = u (\frac{z + \bar{z}}{2}, \frac{z - \bar{z}}{2 i}) + i v (\frac{z + \bar{z}}{2}, \frac{z - \bar{z}}{2 i})$ .

This is an identity even when $x$ and $y$ are not real, i.e. the two variables $z$ and $\bar{z}$ may be considered independent. Putting $\bar{z} = z$ we get $f (z) = u (z, 0) + i v (z, 0)$ .

Given only real part

Problem: $u (x, y)$ is known, $v (x, y)$ is unknown, $f (x + i 0)$ is real; what is $f (z)$ ?

Answer: $f (z) = u (z, 0)$ .

Only example 1 applies here: with $u (x, y) = x^{4} - 6 x^{2} y^{2} + y^{4}$ we obtain $f (z) = z^{4}$ .

Proof: " $f (x + i 0)$ is real" means $v (x, 0) = 0$ . In this case the answer to problem 1 becomes $f (z) = u (z, 0)$ .

General version

Problem: $u (x, y)$ is known, $v (x, y)$ is unknown; what is $f (z)$ ?

Answer: $f (z) = u (z, 0) - i \int u_{y} (z, 0) d z$ (where $u_{y} (x, y)$ is the partial derivative of $u (x, y)$ with respect to $y$ ).

Example 1: with $u (x, y) = x^{4} - 6 x^{2} y^{2} + y^{4}$ and $u_{y} (x, y) = - 12 x^{2} y + 4 y^{3}$ we obtain $f (z) = z^{4} + i C$ with real but undetermined $C$ .

Example 2: with $u (x, y) = \cos (x) \exp (- y)$ and $u_{y} (x, y) = - \cos (x) \exp (- y)$ we obtain $f (z) = \cos (z) + i \int \cos (z) d z = \cos (z) + i (\sin (z) + C) = \exp (i z) + i C$ .

Proof: This follows from $f (z) = u (z, 0) + i \int v_{x} (z, 0) d z$ and the 2nd Cauchy-Riemann equation $u_{y} (x, y) = - v_{x} (x, y)$ .

Given only imaginary part

Problem: $u (x, y)$ is unknown, $v (x, y)$ is known; what is $f (z)$ ?

Answer: $f (z) = \int v_{y} (z, 0) d z + i v (z, 0)$ .

Example 1: with $v (x, y) = 4 x^{3} y - 4 x y^{3}$ and $v_{y} (x, y) = 4 x^{3} - 12 x y^{2}$ we obtain $f (z) = \int 4 z^{3} d z + i 0 = z^{4} + C$ with real but undetermined $C$ .

Example 2: with $v (x, y) = \sin (x) \exp (- y)$ and $v_{y} (x, y) = - \sin (x) \exp (- y)$ we obtain $f (z) = - \int \sin (z) d z + i \sin (z) = \cos (z) + C + i \sin (z) = \exp (i z) + C$ .

Proof: This follows from $f (z) = \int u_{x} (z, 0) d z + i v (z, 0)$ and the 1st Cauchy-Riemann equation $u_{x} (x, y) = v_{y} (x, y)$ .

References

1 2 Milne-Thomson, L. M. (July 1937). "1243. On the relation of an analytic function of z to its real and imaginary parts". The Mathematical Gazette. 21 (244): 228–229. doi:10.2307/3605404. JSTOR 3605404. S2CID 125681848.

[TMG-1] 1 2 Milne-Thomson, L. M. (July 1937). "1243. On the relation of an analytic function of z to its real and imaginary parts". The Mathematical Gazette. 21 (244): 228–229. doi:10.2307/3605404. JSTOR 3605404. S2CID 125681848.

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