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

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Let   and   where   and   are real.

Let   be any holomorphic function.

Example 1:  

Example 2:  

In his article,[1] Milne-Thomson considers the problem of finding   when 1.   and   are given, 2.   is given and   is real on the real axis, 3. only   is given, 4. only   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.

1st problem

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Problem:   and   are known; what is  ?

Answer:  

In words: the holomorphic function   can be obtained by putting   and   in  .

Example 1: with   and   we obtain  .

Example 2: with   and   we obtain  .

Proof:

From the first pair of definitions   and  .

Therefore  .

This is an identity even when   and   are not real, i.e. the two variables   and   may be considered independent. Putting   we get  .

2nd problem

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Problem:   is known,   is unknown,   is real; what is  ?

Answer:  .

Only example 1 applies here: with   we obtain  .

Proof: "  is real" means  . In this case the answer to problem 1 becomes  .

3rd problem

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Problem:   is known,   is unknown; what is  ?

Answer:   (where   is the partial derivative of   with respect to  ).

Example 1: with   and   we obtain   with real but undetermined  .

Example 2: with   and   we obtain  .

Proof: This follows from   and the 2nd Cauchy-Riemann equation  .

4th problem

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Problem:   is unknown,   is known; what is  ?

Answer:  .

Example 1: with   and   we obtain   with real but undetermined  .

Example 2: with   and   we obtain  .

Proof: This follows from   and the 1st Cauchy-Riemann equation  .

References

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  1. ^ a b 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.