Are these functions analytic?

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In summary, if f0 = u(x,y) + i*v(x,y) is analytic in domain D, then f1 = (u^2 - v^2) - 2i*u*v is also analytic in domain D. Similarly, f2 = (e^u)cos(v) + i*(e^u)sin(v) is also analytic in domain D if u and v are continuous and differentiable on domain D. However, f3 = u - i*v is not analytic in domain D as it is the conjugate of f0, which is not differentiable everywhere.
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hadroneater
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Homework Statement


Given that f0 = u(x,y) + i*v(x,y) is analytic in domain D, are these functions analytic in domain D?
1. f1 = (u^2 - v^2) - 2i*u*v
2. f2 = (e^u)cos(v) + i*(e^u)sin(v)
3. f3 = u - i*v


Homework Equations


cauchy-riemann equations

The Attempt at a Solution


I kind of have an idea but I want to double check if my reasoning is right.

If f0 is analytic then all of the partial derivatives of u(x,y) and v(x,y) wrt x and y exist and are continuous on domain D.

1. f1 = f0^2. The product of differentiable and continuous function is also differentiable and continuous on domain D. Therefore f1 is analytic on domain D.

2. f2 = e^u * 0.5*(e^-v + e^v) + i * e^u * 0.5(e^-v - e^v)
Unsure. I can't explicity use Cauchy-Riemann to check. But I know that v and u are continuous. If a function g(x) is continuous and differentiable on some domain or set, is h(x) = e^g(x) also? If so, then f2 is analytic.

3. f3 = conjugate(f0). Not differentiable because conjugate(z) is not differentiable everywhere. Therefore not analytic.
 
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  • #2
I agree with all of those. h(x)=e^g(x) is analytic if g(x) is analytic. It's a composition of analytic functions. If f(z) is analytic and g(z) is analytic then f(g(z)) is analytic.
 

1. What does it mean for a function to be analytic?

An analytic function is a mathematical function that can be expressed as a power series of x. This means that the function has a well-defined derivative at every point in its domain.

2. How can I tell if a function is analytic?

A function is analytic if it can be written as an infinite sum of powers of x. This means that the function is continuous, differentiable, and has a convergent Taylor series at every point in its domain.

3. Can a function be analytic at some points but not others?

Yes, it is possible for a function to be analytic at some points but not others. For example, a function may have a singularity or discontinuity at a certain point, making it non-analytic at that point.

4. Are all continuous functions analytic?

No, not all continuous functions are analytic. A function must have a convergent Taylor series at every point in its domain to be considered analytic.

5. What is the significance of a function being analytic?

Analytic functions have many important properties that make them useful in mathematical and scientific applications. They can be easily approximated by polynomials and have well-defined derivatives, making them useful for calculating rates of change and optimizing functions. In addition, many important mathematical concepts, such as complex numbers, are based on analytic functions.

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