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Milne-Thomson method for finding a holomorphic function

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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.