1. Show that if u is harmonic in a domain Ω, then also
the derivatives of...
1. Show that if u is harmonic in a domain Ω, then also
the derivatives of
u of any order are harmonic in Ω. (Hint: To get the result for any
order
you may want to use induction).
For the following u(x, y), show that it is harmonic and then
find a corresponding v(x, y) such that f(z)=u+iv is analytic.
u(x, y)=(x^2-y^2) cos(y)e^x-2xysin(y)ex
Suppose that Ω ⊆ R n is bounded, and path-connected, and u ∈
C2 (Ω) ∩ C(∂Ω) satisfies ( −∆u = 0 in Ω, u = g on ∂Ω.
Prove that if g ∈ C(∂Ω) with g(x) = ( ≥ 0 for all x ∈ ∂Ω, > 0
for some x ∈ ∂Ω, then u(x) > 0 for all x ∈ Ω
Let u(x, y) be the harmonic function in the unit disk with the
boundary values u(x, y) = x^2 on {x^2 + y^2 = 1}. Find its
Rayleigh–Ritz approximation of the form x^2 +C1*(1−x^2
−y^2).
Superposition of N harmonic oscillator waves of equal amplitude
equal angular frequency ω and constant incremental phase difference
φ. And constant spacing d between them. The total length of the
array of the oscillator is L. With L = N*d
The amplitude is: where A0 the amplitude of each wave. A = A0
sin(Nφ /2) / sin(φ /2)
The intensity : I = I0*(sin(Nφ /2) / sin(φ /2))^2
1. show the minimum is at Nd*sin(θ ) = m λ, m...
1.- Show that (R, τs) is connected. Also show that (a, b) is
connected, with the subspace topology given by τs.
2. Let f: X → Y continue. We say that f is open if it sends open
of X in open of Y. Show that the canonical projection
ρi: X1 × X2 → Xi
(x1, x2) −→ xi
It is continuous and open, for i = 1, 2, where (X1, τ1) and (X2,
τ2) are two topological spaces and...
If the function u (x, y) is a harmonic conjugate of v (x, y) prove that the curves u (x, y) = st. and v (x, y) = stations. are orthogonal to each other. These curves are called level curves. Now consider the function f (z) = 1 / z
defined throughout the complex plane except the beginning of the axes. Draw them
level curves for the real and imaginary part of this function
and notice that they are two...
Find u(x,y) harmonic in S with given boundary values:
S = {(x,y): 1 < y < 3} , u(x,y) = 5 (if y=1) and = 7
(when y=3)
S = {(x,y): 1 < x2 + y2 < 4},
u(x,y)= 5 (on outer circle) and = 7 (on inner circle)
I have these two problems to solve, and I'm not sure where to
start. Any help would be appreciated. Thanks!