A)Let a be less than b and set M(z)=(z-ia)/(z-ib). Define the lines L1={z:F(z)=b},
L2={z:F(z)=a} and L3={z:R(z)=0}. The three lines split the complex plane into 6 regions. Determine the image of them in the complex plane.

b) Let log be principal branch of the logarithm. Show that log(M(z)) is defined for all z in C with the exception of the linesegment from ia to ib.

c) Define h(z)=F(log(M(z))) for R(z)>0. Show that h is harmonic and that h(z) is greater than 0 and less than pi

d)Show that log(z-ic) is defined for R(z)>0 and any real number c. Prove that
|F(log(z-ic))|is less than pi/2 in this region

e) Prove that h(z)=F(log(z-ia) - log(z-ib))

f) Use the fundamental theorem of calculus to show that integral(from a to b) of
dt/z-it = i(log(z-ib)-log(z-ia))

g) Combine part e and f to show that
h(x+iy)=integral(from a to b) of xdt/(x^2 +(y-t)^2)=arctan((y-a)/x)-arctan((y-b)/x)

h)Intrpret part g geometrically by showing that h(z) measures the interior angle of the triangle with vertices ia, ib and z at the vertex z. What are the limits of h(z) as
R(z)->0 for F(z) in (a,b) and for F(z) not in [a,b]?

Solution Summary

Harmonic functions are investigated. The solution is detailed and well presented.

...Harmonic oscillator eigen functions are a function of Hermit polynomials. ...Harmonic oscillator eigen functions are a function of Hermit polynomials. ...

... b). Now we have harmonic oscillator as unperturbed system and delta-function potential as perturbation. ... Wave function of quantum harmonic oscillator (see Ref. ...

... K of the block as a function of time. ... answers to various questions related to harmonic oscillator equations ... major characteristics of motion as functions of time ...

... operators on the eigenstates of the harmonic oscillator are ... see that the integrand is an odd function (a product of an odd and even functions) so integrating ...

... Therefore, the system will undergo simple harmonic motion. ... x(t)= Acos(sqrt k/mt). a). Find the velocity v of the block as a function of time. ...

... to consider the case of N identical particles in the same harmonic potential. ... but if we take the N-th power of the single particle partition function we count ...

... to these two energy levels in terms of harmonic oscillator wave functions and the ... Where is the nth wave function of the oscillator and is the part ...