1. ## harmonic functions

Suppose that the functions u and v are harmonic in a domain D:
(a) Is the product uv necessarily harmonic in D?
(b) Is du/dx harmonic in D? (You may use the fact that harmonic functions have continuous
partial derivatives of all orders.)

In the previous question i have shown that if v is harmonic conjugate for u in a domain D, then uv is harmonic in D.
Now i'm assuming that uv is only harmonic in D if v is harmonic conjugate for u, but can u and v both be harmonic without being conjugate of the other? If so, can someone tell me how to explain it in words in a general sense without actually being given specific values for u or v.

For b) i think it will be harmonic if uxxx + uxyy = 0. Can we know if this is the case given we know uxx and uyy = 0?

Any help would be appreciated this is a very confusing topic for me. Thanks

2. ## Re: harmonic functions

Originally Posted by linalg123
Suppose that the functions u and v are harmonic in a domain D:
(a) Is the product uv necessarily harmonic in D?
(b) Is du/dx harmonic in D? (You may use the fact that harmonic functions have continuous
partial derivatives of all orders.)

In the previous question i have shown that if v is harmonic conjugate for u in a domain D, then uv is harmonic in D.
Now i'm assuming that uv is only harmonic in D if v is harmonic conjugate for u, but can u and v both be harmonic without being conjugate of the other? If so, can someone tell me how to explain it in words in a general sense without actually being given specific values for u or v.

For b) i think it will be harmonic if uxxx + uxyy = 0.
What does 'it' refer to here? You have three functions, u, v, and uv. Which are you refering to?

Can we know if this is the case given we know uxx and uyy = 0?
Only if you mean "identically equal to 0" which you do not here. You do not even know that uxx and uyy, individually, at any specific point. What definition of "harmonic" are you using?

Any help would be appreciated this is a very confusing topic for me. Thanks

3. ## Re: harmonic functions

ok i guess i didn't state my questions clearly enough, i'm sorry.
PART A
What i know: If i let m=uv, mxx + myy = (uxx + uyy)v + 2(uxvx + uyvy) + u(vxx+vyy)
Okay. Now we know that u and v are harmonic, so the first and third terms are both 0.
However, I believe the middle term will only disappear if u and v satisfy the cauchy-riemann equations?

Question: Can the functions u and v both be harmonic in D, but NOT satisfy the cauchy riemann equations?

PART B
we know u is harmonic in D.
i.e uxx+uyy= 0

we need to know if this means ux is also harmonic in D, i.e (ux)xx + (ux)yy = 0
we are given that harmonic functions have continuous partial derivatives of all orders.

Question: Do we have enough here to say whether ux is harmonic in D?