Harmonic Functions

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Harmonic Functions
MTH 324
Lecture 13
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Previous Lectures Review

• Essential condition for a function to be
  analytic
• Cauchy-Riemann equations in polar coordinates
• Sufficient condition for analyticity using
  polar
• coordinates

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Lectures Outline

• Second order partial derivatives
• Laplace equation
• Harmonic functions
• Harmonic conjugate
• construction of analytic function

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Second order partial derivatives
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Two dimensional Laplace equation
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Harmonic function
Remark
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Example
solution
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Theorem
Proof.
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Example
Solution
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Example
Solution
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Harmonic conjugate functions
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Example
Solution
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Example
Solution
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Direct method of constructing an analytic
function
Working Rules
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Example
Solution
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Harmonic equation in polar form
Example
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Solution
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References

• A First Course in Complex Analysis with
  Applications by Dennis G. Zill and Patrick D.
  Shanahan.
• Complex variables and applications by James
  Brown and Ruel Churchill
• Fundamentals of complex Analysis by Muhammad Iqbal