Complex analysis: Harmonic functions

This lecture is part of an online undergraduate course on complex analysis.

We study the question: when is a function u the real part of a holomorphic function w=u+iv? An easy necessary condition is that u mist be harmonic. We use the Caucy-Riemann equations to show that this condition is also sufficient if u is defined in a simply connected open set, and given an example to show that it need not be sufficient if u is defined on a non-simply connected set. So on simply connected open sets harmonic functions are the same as the real parts of holomorphic functions.

[thomasjefferson]

Not gonna lie, I'm enjoying this dudes lectures

[omarhayat]

thank you for these lectures on complex analysis! Merry Christmas!

[lefximusic]

While watching this video, I have several problems going through the lecture
First I don’t really know the exact definition of all these connected sets, which professor have been mentioning quite regularly
And during the explanation of why we cannot extend u to a holomorphic functions in a region with a hole
Can someone explain these to me? With full appreciation

[robertfulton]

I'm having trouble understanding why the space of homogeneous harmonic polynomials of degree d > 0 is two-dimensional (5:13). Can anybody explain why this is?

[lilbthebasegod]

25:18 I think the compactness of [0, 1] is necessary to guarantee that v is constant.. is it right?