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 x and y are real.
Hence,
![{\displaystyle {\begin{aligned}x&={\frac {z+{\bar {z}}}{2}}\\[6pt]{\text{and }}y&={\frac {z-{\bar {z}}}{2i}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8faa56b9b450221d7566addbbb4e055169647df5)
Therefore,
is equal to
![{\displaystyle f(z)=u\left({\frac {z+{\bar {z}}}{2}}\ ,{\frac {z-{\bar {z}}}{2i}}\right)+iv\left({\frac {z+{\bar {z}}}{2}}\ ,{\frac {z-{\bar {z}}}{2i}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7af0694e69f23c76fe15f368cfb6aff7d53d85)
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
![{\displaystyle f'(z)={\partial u \over \partial x}-i{\partial u \over \partial y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48b519e2cfeb1b2f5e216614ff33063ef069df40)
![{\displaystyle f'(z)=\Phi (x,y)-i\Psi (x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/854b3561ebbc3578ae548f0a4841b4e1cb441bb7)
Now, putting
and
in the above equation, we get
![{\displaystyle f'(z)=\Phi (z,0)-i\Psi (z,0).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3d39e1c64ac725f31bbe4c97ef1dde1c108c5e)
Integrating both sides of the above equation we get
![{\displaystyle \int f'(z)\,dz=\int \Phi (z,0)\,dz-i\int \Psi (z,0)\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ac55ae0f6686420866a322962953b500820b6f)
or
![{\displaystyle f(z)=\int f'(z)\,dz=\int \Phi (z,0)\,dz-i\int \Psi (z,0)\,dz+c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/275ee4a66ad4e7b333d1a0f3b87a1aef3e4d5979)
which is the required holomorphic function.
Example
Let
, and let the desired holomorphic function be
Then as per the above process we know that
![{\displaystyle f'(z)={\partial u(x,y) \over \partial x}+i{\partial v(x,y) \over \partial x}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15fa36b48578873b321c6b821442a8ed0ba23e3c)
But as
is holomorphic, so it satisfies Cauchy–Riemann equations.
Hence,
and
Or
and
.
Substituting these values in
we get,
![{\displaystyle f'(z)={\partial u(x,y) \over \partial x}-i{\partial u(x,y) \over \partial y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7c2be2e525fdc3a562102cdda5ccfd213bf2cb)
Hence,
![{\displaystyle f'(z)=(4x^{3}-12xy^{2})-i(-12x^{2}y+4y^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf9c665a9f34d40bb9a0f07ea9e727f45129d16)
This can be written as
where,
and
.
Rewriting
using
and
![{\displaystyle f'(z)=4z^{3}-i(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95c97d5c6bad794b62b5103671cf1dc760b2d8f0)
Integrating both sides w.r.t
we get,
![{\displaystyle \int f'(z)\,dz=\int 4z^{3}dz+\int 0\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f537cd125353fc127a7abfbbdbfe7d854ab37afc)
Hence,
is the required holomorphic function.
References
- ^ 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. doi:10.2307/3605404. JSTOR 3605404.