1. Method for finding the holomorphic function

2. Example

3. References

In mathematics, the Milne-Thomson method is a method of finding a holomorphic function, whose real or imaginary part is given.^[1] The method greatly simplifies the process of finding the holomorphic function whose real or imaginary or any combination of the two parts is given. It is named after Louis Melville Milne-Thomson.

Method for finding the holomorphic function

Let be any holomorphic function.

Let and where and are real.

Hence,

Therefore, is equal to

This can be regarded as an identity in two independent variables and . We can therefore, put and get

So, can be obtained in terms of simply by putting and in when is a holomorphic function.

Now, .

Since, is holomorphic, hence Cauchy–Riemann equations are satisfied. Hence, .

Let and .

Then

Now, putting and in the above equation, we get

Integrating both sides of the above equation we get

or

which is the required holomorphic function.

Example

Let , and let the desired holomorphic function be

Then as per the above process we know that

But as is holomorphic, so it satisfies Cauchy–Riemann equations.

Hence, and

Or and .

Substituting these values in we get,

Hence,

This can be written as

where, and .

Rewriting using and

Integrating both sides w.r.t we get,

Hence, is the required holomorphic function.

References

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