Analytic Function (3+x-iy)^7 in Domain [z]<2?

  • Context: Graduate 
  • Thread starter Thread starter tibphysic
  • Start date Start date
  • Tags Tags
    Functions
Click For Summary
SUMMARY

The function f(z) = (3+x-iy)^7 is not analytic in the domain |z| < 2. This conclusion is drawn from the calculation of the derivative with respect to the conjugate variable, yielding df/dz̅ = 7(3+z̅)^6, which is not equal to zero for |z| < 2. Therefore, the function fails to satisfy the Cauchy-Riemann equations necessary for analyticity in the specified domain.

PREREQUISITES
  • Understanding of complex variables and functions
  • Familiarity with the Cauchy-Riemann equations
  • Knowledge of complex differentiation
  • Concept of analytic functions in complex analysis
NEXT STEPS
  • Study the Cauchy-Riemann equations in detail
  • Learn about complex differentiation techniques
  • Explore the properties of analytic functions
  • Investigate the implications of non-analytic functions in complex analysis
USEFUL FOR

Students and professionals in mathematics, particularly those specializing in complex analysis, as well as researchers exploring properties of analytic functions and their applications.

tibphysic
Messages
3
Reaction score
0
(3+x-iy)^7 is analytic function of the complex variable z=x+iy in the domain [z]<2?
 
Physics news on Phys.org
Nope, [tex]f(z) = (3+x-iy)^7 = (3+\bar{z})^7[/tex]

[tex]\frac{df}{d\bar{z}} = 7(3+\bar{z})^6 \neq 0, \,\mathrm{for}\; |z| < 2.[/tex]
 

Similar threads

Undergrad How to see that cos(z) is unbounded, but cos(xy)+isin(xy) is bounded?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad How can you represent a point by "z = x + iy" as shown here?
  • · Replies 10 ·
Replies
10
Views
2K
Graduate Hopf fibration of 3-sphere
  • · Replies 39 ·
2
Replies
39
Views
5K
Undergrad Finding the inverse tangent of a complex number
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad How do I separate the real and imaginary parts of an infinite product?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
3K
High School Continuity of ln(x) function
  • · Replies 11 ·
Replies
11
Views
3K
Is f(z) =(1+z)/(1-z) a real function?
  • · Replies 17 ·
Replies
17
Views
4K
MHB Image of the upper half complex plane, under the function g(z) = exp(2 \pi i z).
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Complex Analysis: Cauchy Riemann Equations 2
  • · Replies 2 ·
Replies
2
Views
2K