Find all the points where [tex]f(z) = (x^2 + y^2 -2y) + i(2x-2xy)[/tex] is differentiable, and compute the derivative at those points.
Is the function above analytic at any point? Justify your answer clearly.
[tex]u (x,y) = x^2 + y^2 - 2y[/tex]
[tex]v (x,y) = 2x - 2xy [/tex]
[tex]u_x = 2x[/tex]
[tex]v_y = -2x[/tex]
[tex]u_y = 2y - 2[/tex]
[tex]v_x = 2 - 2y[/tex]
However Cauchy-Riemann states that [tex]u_x = v_y[/tex] so my reasoning is [tex]v_y = -v_y[/tex] and that is only true where [tex]v_y = 0[/tex]. That is to say: [tex]-2x = 0 \rightarrow x = 0[/tex].
But if [tex]x=0[/tex] then [tex]v(x,y) = 0[/tex] and [tex]u(x,y) = y^2 - y[/tex]
We then continue: By Cauchy-Riemann:
[tex]u_y = -v_x[/tex]
But if [tex]v(x,y) = 0[/tex] then [tex]v_x = 0[/tex]
And as such: [tex]2y - 2 = 0[/tex]
[tex]y = 1[/tex]
Does this mean the function is only differentiable at (0,1) ?
The derivative of the function:
[tex]f'(z_0) = u_x + iv_x = 2x + i(2-2y)[/tex]
At the point (0,1):
[tex]f'(z_0) = 0 + i (2-2) = 0[/tex]
[itex]f'(z_0)= 2x+ i(2- 2y)[/itex]
snipez90 said:You must compute the Cauchy-Riemann equations first, then look at the set of (x,y) that satisfy the equations. Then you can determine where the function is analytic (by HallsofIvy's given definition), if anywhere.
The Cauchy-Riemann equations, named after mathematicians Augustin Cauchy and Bernhard Riemann, are a set of two partial differential equations that describe the relationship between the real and imaginary parts of a complex-valued function. They are an important tool in the study of complex analysis and are used to determine if a function is analytic or not.
The Cauchy-Riemann equations are significant because they provide a necessary and sufficient condition for a function to be analytic. This means that if a function satisfies the Cauchy-Riemann equations, it is guaranteed to be analytic. Analytic functions have many useful properties, including the ability to be differentiated and integrated along any path in the complex plane.
The Cauchy-Riemann equations are closely related to the concept of holomorphicity, which is a key property of complex-valued functions. A function is said to be holomorphic if it is analytic in a given region of the complex plane. The Cauchy-Riemann equations provide a way to check if a function is holomorphic, as they are a necessary and sufficient condition for analyticity.
Yes, the Cauchy-Riemann equations can be used to solve complex integration problems. This is because they allow us to determine if a function is analytic, and if it is, we can use techniques from complex analysis to integrate the function along any path in the complex plane. This is known as the Cauchy integral theorem and is a powerful tool in solving complex integration problems.
Yes, the Cauchy-Riemann equations have many real-world applications. They are used in fields such as engineering, physics, and economics to model and analyze systems that involve complex variables. They are also used in the study of fluid dynamics, as they can be used to describe the flow of fluids in two dimensions. Additionally, the Cauchy-Riemann equations have applications in image processing, where they can be used to analyze and manipulate digital images.