Find An Analytic Function Whose Imaginary Part Is Given | How to find an analytic function | #maths

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Welcome FUTURE IITians,
This YouTube video title explores a fundamental concept in Complex Analysis: finding an analytic function when its real part is given. Here's a detailed overview:

What are Analytic Functions?
Analytic functions are complex-valued functions that are differentiable at every point in their domain. They're crucial in Complex Analysis, as they enable the application of powerful theorems like Cauchy's Integral Formula and Residue Theorem.

✳️🔅✳️Finding an Analytic Function from its Imaginary Part-:
Given a real-valued function u(x,y), we can find an analytic function f(z) = u(x,y) + iv(x,y) using various methods, including:

(1) Cauchy-Riemann Equations -: These equations relate the partial derivatives of u and v, enabling us to find v(x,y) and construct f(z).
(2) Milne-Thomson Method -: This method involves using the Cauchy-Riemann equations to find v(x,y) and construct f(z).

✳️🔅✳️Step-by-Step Process -:
To find an analytic function from its imaginary part, follow these steps:

1.Verify the given real part u(x,y) satisfies Laplace's equation*: ∂²u/∂x² + ∂²u/∂y² = 0.
2.Use the Cauchy-Riemann equations to find v(x,y)*: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.
3.Integrate the Cauchy-Riemann equations to find v(x,y)*: Use the equations to find v(x,y) up to an additive constant.
4.Construct the analytic function f(z)*: Combine u(x,y) and v(x,y) to form f(z) = u(x,y) + iv(x,y).

✳️🔅✳️Example Problems -:
Some example problems might include:

- Finding an analytic function f(z) given u(x,y) = x² - y².
- Determining v(x,y) given u(x,y) = e^x cos(y).

✳️🔅✳️Importance in Complex Analysis -:
Understanding how to find an analytic function from its real part is essential in Complex Analysis, as it enables the application of various theorems and techniques, such as contour integration and residue theory.

✳️🔅✳️Tips and Tricks-:
Use the Cauchy-Riemann equations to relate u and v -: These equations are fundamental in finding v(x,y) and constructing f(z).
Verify the given real part satisfies Laplace's equation -:This ensures the existence of an analytic function.

By mastering this concept, you'll be able to tackle complex problems in Complex Analysis and apply these techniques to various fields, such as physics, engineering and mathematics.
✳️🔅✳️ HASHTAGS -:
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✳️🔅✳️ Method we used in the given problem is -:
"Milne-Thomson Method"
The Milne-Thomson method is a technique used to find an analytic function f(z) = u(x,y) + iv(x,y) when its real part u(x,y) is given.

Steps -:
1.Express u(x,y) in terms of z and conjugate(z) : Replace x and y with z = x + iy and conjugate(z) = x - iy.
2.Find the partial derivatives : ∂u/∂z and ∂u/∂(conjugate(z)).
3.Use the formula : f'(z) = ∂u/∂z.
4.Integrate f'(z) : Find f(z) by integrating f'(z) with respect to z.

Example -:
Suppose u(x,y) = x^2 - y^2. To find f(z):

1.Express u in terms of z and conjugate(z) : u = (z + conjugate(z))^2/4 - (z - conjugate(z))^2/(-4) = (z^2 + conjugate(z)^2)/2.
2.Find the partial derivatives : ∂u/∂z = z.
3.Use the formula : f'(z) = z.
4.Integrate f'(z) : f(z) = ∫ z dz = z^2/2 + C.

The Milne-Thomson method provides a powerful technique for finding analytic functions from their real parts.

✳️🔅✳️Would you like more examples or details?
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