Purely Real-valued Analytic Functions?

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In summary: However, functions like sin(x) and cos(x) are complex analytic functions, since they satisfy the CR conditions in the complex plane. In summary, a function which is purely real-valued cannot be analytic unless it is a constant. Functions like sin(x) and cos(x) are complex analytic functions and satisfy the CR conditions in the complex plane.
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Homework Statement


Can a function which is purely real-valued be analytic? Describe the behavior of such functions?


Homework Equations


The Cauchy-Riemann conditions
ux=vy, vx=-uy

The Attempt at a Solution



I can't think of any pure real-valued equations off the top of my head which satisfy the CR conditions. However, could any function like sin(x), cos(x), e^x, which is infinitely differentiable be considered analytic? Doesn't infinite differentiability imply that the CR conditions are already satisfied?
 
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  • #2
jtleafs33 said:

Homework Statement


Can a function which is purely real-valued be analytic? Describe the behavior of such functions?


Homework Equations


The Cauchy-Riemann conditions
ux=vy, vx=-uy

The Attempt at a Solution



I can't think of any pure real-valued equations off the top of my head which satisfy the CR conditions. However, could any function like sin(x), cos(x), e^x, which is infinitely differentiable be considered analytic? Doesn't infinite differentiability imply that the CR conditions are already satisfied?

No. If your function is real valued then v=0. What does that tell you about u?
 
  • #3
If v=0, then vx=vy=0. So then, u must be constant, so that it's derivative will also be zero, and thus satisfy the CR conditions?
 
  • #4
jtleafs33 said:
If v=0, then vx=vy=0. So then, u must be constant, so that it's derivative will also be zero, and thus satisfy the CR conditions?

Yes, that's it. The only purely real analytic functions are constant.
 

1. What are purely real-valued analytic functions?

Purely real-valued analytic functions are functions that can be represented as a power series with real coefficients, meaning that the function is defined for all real numbers and its derivatives can be calculated at every point.

2. What is the difference between a real-valued analytic function and a purely real-valued analytic function?

A real-valued analytic function can have complex coefficients in its power series representation, while a purely real-valued analytic function has only real coefficients.

3. What is the importance of purely real-valued analytic functions in mathematics?

Purely real-valued analytic functions have many applications in mathematics, particularly in areas such as complex analysis, differential equations, and Fourier analysis. They also have connections to other branches of mathematics, such as number theory and algebraic geometry.

4. How can one determine if a function is purely real-valued analytic?

A function can be shown to be purely real-valued analytic if it can be written as a power series with real coefficients and if its derivatives exist at every point within its domain.

5. Are there any well-known examples of purely real-valued analytic functions?

Yes, some well-known examples include the exponential function, trigonometric functions, and polynomial functions. In fact, any function that can be represented as a power series with real coefficients is a purely real-valued analytic function.

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