Park City Mathematics Institute
Undergraduate Summer School 2018
Introduction to Harmonic Analysis
- Let be a rotation in the plane.
- Consider the change of variables . Then .
- If is harmonic, then is also harmonic.
- Let be the polar coordinates of the plane. Then
- Let be a harmonic function on . Then there exists a harmonic function that is conjugate to , so is holomorphic. (Hint: Consider a line integral of the 1-form .)
- If and are conjugate to in the plane, then is a constant.
- If 0 is conjugate to in the plane, then is a constant.
- If is holomorphic in and real valued, then is a constant.
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