What is Milne-Thomson Method?

Welcome back MechanicaLEi, did you know that Milne-Thompson Method was developed by L. M. Milne-Thomson and helped greatly simplify the process of finding a holomorphic function whose real or imaginary or any combination of the two parts is given? This makes us wonder, what is Milne-Thomson Method? Before we jump in check out the previous part of this series to learn about what analytic functions are? Now, Milne-Thomson Method is used to construct an analytic function when its real and or imaginary components are known. If a given function f of z is analytic in a given domain then f of z can be integrated in the domain using anti-derivative, that is by find a capital f of z such that capital f dash of z equals f of z. Let's look at an example to understand better. Consider u equal to the following expression. we need to find f of z by Milne-Thomson method. Following the method, we first find partial derivative of u with respect to x called ux and then find partial derivative of u with respect to y called uy. Next, we substitute y equals to zero in both ux and uy to get psi one of x comma zero and psi two of x comma zero. After finding these values we simply substitute z in place of x in both psi one and psi 2 and write it in the form of anti-derivative f dash of z. We then integrate this anti-derivative to find f of z and obtain the solution. Hence, we first saw what Milne-Thomson Method is and then went on see an example of it?

In the next episode of MechanicaLEi find out what Harmonic functions are?

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