Verifying Entireness of Analytic Functions Using Cauchy-Riemann Theory

  • Thread starter Benzoate
  • Start date
  • Tags
    Function
In summary, to verify that the function f(z)=3*x+y+i(3y-x) is entire, we can apply the rules of Cauchy-Riemann theory to show that it satisfies the criteria for being analytic at each point. This can be done by checking that the partial derivatives of the real and imaginary parts of the function satisfy the Cauchy-Riemann equations u_x=v_y and u_y=-v_x. If these equations are satisfied everywhere, then the function is entire. Additionally, we need to check for any singularities that could prevent the function from being entire.
  • #1
Benzoate
422
0

Homework Statement


Apply rules of Cauchy-riemann theory to verify that each of these functions is entire:

f(z)=3*x+y+i(3y-x)



Homework Equations



u_x=v_y, u_y=-v_x

The Attempt at a Solution



u(x,y)=3x+y
v(x,y)=3y-x

u_x=3
v_y=3
u_y=1
-v_y=1
I know that a function is analytic at each point, then the function is entire. How would I show that the function is analytic at each pt.?
 
Physics news on Phys.org
  • #2
You've shown it satisfies CR everywhere. The only thing that could prevent it from being entire is if it has a removable singularity someplace. Does it? Does it have ANY singularities?
 

FAQ: Verifying Entireness of Analytic Functions Using Cauchy-Riemann Theory

What is an analytic function problem?

An analytic function problem is a mathematical problem that involves finding a function that satisfies a given set of conditions or constraints. These problems often involve complex numbers and are commonly encountered in fields such as engineering, physics, and economics.

What are some common techniques for solving analytic function problems?

Some common techniques for solving analytic function problems include the use of calculus, differential equations, and complex analysis. These techniques allow for the manipulation and analysis of functions to find solutions that satisfy the given conditions.

Can analytic function problems have multiple solutions?

Yes, analytic function problems can have multiple solutions. It is common for these problems to have an infinite number of solutions, particularly when working with complex numbers.

How are analytic function problems used in real-world applications?

Analytic function problems are used in a wide range of real-world applications, including signal processing, control systems, and optimization. These problems are particularly useful in situations where a precise mathematical model is needed to describe a system or process.

What are some common challenges when solving analytic function problems?

Some common challenges when solving analytic function problems include understanding and interpreting the given conditions, selecting an appropriate technique for solving the problem, and dealing with the complexity of functions involving complex numbers.

Similar threads

Is this complex function analytic?
Replies
14
Views
2K
The Cauchy Riemann Equations for this Function
Replies
7
Views
1K
Is the Function Analytic? Testing the Cauchy Riemann Equations
Replies
2
Views
1K
Use C-R eqns to determine points whose fn's are analytic
Replies
6
Views
2K
Show that the given function is continuous
Replies
27
Views
2K
Cauchy-Riemann Condition Verification for f(z) = z^3-5iz+√7
Replies
11
Views
2K
Proving Entirety of conj(f(conj(z))) for an Entire Function f
Replies
3
Views
2K
What Constant Makes a Function Holomorphic?
Replies
4
Views
2K
Determining At Which Points A Function is Analytic (Holomorphic) and Singular
Replies
2
Views
6K
Show that f is entire when u is harmonic and v=
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top