Can you please tell me if this question is Correct?

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Discussion Overview

The discussion revolves around the question of whether the function u(x,y) = e3x((3y-2)cos(3y)-2xsin(3y)) is holomorphic in the complex plane C, and whether it can be associated with a complex potential f(z) such that u(x,y) = Re(f(x+iy)). Participants explore the properties of the function, particularly its harmonicity and the implications of Laplace's equation.

Discussion Character

  • Debate/contested, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants propose that the function should be harmonic rather than holomorphic, as a real-valued function cannot be holomorphic unless it is constant.
  • There is a suggestion that the question should focus on whether u(x,y) is harmonic, not holomorphic.
  • Concerns are raised about the correctness of the function itself, with references to Laplace's equation not holding for the given function.
  • One participant suggests that there may be a misprint in the function, proposing that the last 2 should be a 3, which would make the function harmonic.
  • Another participant agrees that the function as presented is not harmonic and confirms the proposed correction.

Areas of Agreement / Disagreement

Participants generally agree that the function is not holomorphic and should be evaluated for harmonicity instead. However, there is uncertainty regarding the correctness of the function itself and whether it contains a misprint.

Contextual Notes

There are unresolved issues regarding the assumptions about the function's form and the implications of Laplace's equation. The discussion reflects a need for clarity on the definitions and properties of harmonic and holomorphic functions.

shen07
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"Show that u(x,y) = e3x((3y-2)cos(3y)-2xsin(3y)) is holomorphic in C.

Hence Find the complex Potential f(z) such that u(x,y)=Re(f(x+iy))."

Well i think it should be harmonic because we cannot show that Re(z) is holomorphic and hence find its Complex Potential, just want you to confirm that with me.
 
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shen07 said:
"Show that u(x,y) = e3x((3y-2)cos(3y)-2xsin(3y)) is holomorphic in C.

Hence Find the complex Potential f(z) such that u(x,y)=Re(f(x+iy))."

Well i think it should be harmonic because we cannot show that Re(z) is holomorphic and hence find its Complex Potential, just want you to confirm that with me.
I agree with you. A real-valued function cannot be holomorphic (unless it is constant). The question should be asking whether u(x,y) is harmonic, not holomorphic.
 
Opalg said:
I agree with you. A real-valued function cannot be holomorphic (unless it is constant). The question should be asking whether u(x,y) is harmonic, not holomorphic.
Ok. Thanks..But the function also is not right then? because LaPlace's Equation does not hold from that function..
 
shen07 said:
"Show that u(x,y) = e3x((3y-2)cos(3y)-2xsin(3y)) is holomorphic in C.

Hence Find the complex Potential f(z) such that u(x,y)=Re(f(x+iy))."

Well i think it should be harmonic because we cannot show that Re(z) is holomorphic and hence find its Complex Potential, just want you to confirm that with me.

shen07 said:
Ok. Thanks..But the function also is not right then? because LaPlace's Equation does not hold from that function..
You're right again! As it stands, that function is not harmonic. Perhaps the last 2 is a misprint for a 3. Then $u(x,y) = e^{3x}\bigl((3y-2)\cos(3y)-\mathbf{3}x\sin(3y)\bigr)$ is harmonic, being the real part of $-(3iz+2)e^{3z}.$
 
Opalg said:
You're right again! As it stands, that function is not harmonic. Perhaps the last 2 is a misprint for a 3. Then $u(x,y) = e^{3x}\bigl((3y-2)\cos(3y)-\mathbf{3}x\sin(3y)\bigr)$ is harmonic, being the real part of $-(3iz+2)e^{3z}.$

Hey Thanks a lot for that, i would have passed hours to find that error, I am really greatful to you.
 

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