Question

Consider the problem of finding a function ϕ that is harmonic in the right half-plane and takes the values ϕ(0,y)=y/(1+y)

) on the imaginary axis. Observe that the obvions first guess
ϕ(z)=Im(z/1-z²),fails because(z/1-z²)
is not analytic at z=1. However, the following strategy can be used.
(a) According to the text, the mappings (7) and (8) provide a correspondence between the right half-plane and the unit disk. (Of course, one should interchange the roles of z and w in the formulas.) Thus the w-plane inherits from ϕ(z) a function ψ(w) harmonic in the unit disk. Show that the values of ψ(w) on the unit circle
w=eᶦθ
must be given by ψ(e) = sinθ/2
​
(b) Argue that the harmonic function ψ(w) must be given by ψ(w)= 1/2Im(w) throughout the unit disk.
(c) Use the mappings to carry ψ(w) back to the z-plane, producing the function
ϕ(z)= y/(y² + (1 +x)²
as a solution of the problem.

Answer 1

To find a function φ that is harmonic in the right half-plane and takes the values φ(0,y)=y/(1+y) on the imaginary axis, we can use the mappings to provide a correspondence between the right half-plane and the unit disk. By inheriting a function ψ(w) that is harmonic in the unit disk, we can determine the values of ψ(w) on the unit circle and prove that the harmonic function ψ(w) is given by a specific expression throughout the unit disk. Finally, we can use the mappings to carry ψ(w) back to the z-plane, obtaining the desired function φ(z) as a solution.

Step-by-step explanation:

To find a function φ that is harmonic in the right half-plane and takes the values φ(0,y)=y/(1+y) on the imaginary axis, we can use the mappings (7) and (8) to provide a correspondence between the right half-plane and the unit disk. We inherit a function ψ(w) that is harmonic in the unit disk from φ(z).

1. To show that the values of ψ(w) on the unit circle w=eiθ are given by ψ(e) = sin(θ/2), we substitute w=eiθ into the function ψ(w) and simplify the expression using Euler's formula and trigonometric identities.
2. To argue that the harmonic function ψ(w) must be given by ψ(w) = 1/2Im(w) throughout the unit disk, we use the fact that harmonic functions satisfy the Cauchy-Riemann equations.
3. Using the mappings, we carry ψ(w) back to the z-plane, producing the function φ(z) = y/(y² + (1 + x)² as a solution to the problem.