1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

4.5 Potential in a Disk 279

principle,whas maximum and minimum values 0, and thereforewis identi-
cally 0 throughoutR.Inotherwords,uandvare identical.

EXERCISES

1. Solve the potential equation in the disk 0<r<cif the boundary condi-
   tion isv(c,θ)=|θ|,−π<θ≤π.


2. Same as Exercise 1 ifv(c,θ)=θ,−π<θ<π. Is the boundary condition
   satisfied atθ=±π?


3. Same as Exercise 1, with boundary condition

v(c,θ)=f(θ )=

{

cos(θ ), −π/ 2 <θ <π/2,

0 , otherwise.

4. Find the value of the solution atr=0 for the problems of Exercises 1, 2,
   and 3.


5. If the functionf(θ )in Eq. (2) is continuous and sectionally smooth and
   satisfiesf(−π+)=f(π−), what can be said about convergence of the
   series forv(c,θ)?


6. Show that

v(r,θ)=a 0 +

∑∞

n= 1

r−n

(

ancos(nθ)+bnsin(nθ)

)

is a solution of Laplace’s equation in the regionr>c(exterior of a disk)

and has the property that|v(r,θ)|is bounded asr→∞.

7. If the conditionv(c,θ)=f(θ )is given, what are the formulas for thea’s
   andb’s in Exercise 6?


8. The solution of Eqs. (1)–(4) can be written in a single formula by the
   following sequence of operations:
   a.Replaceθbyφin Eq. (11) for thea’s andb’s;
   b.replace thea’s andb’s in Eq. (10) by the integrals in parta;
   c. use the trigonometric identity

cos(nθ)cos(nφ)+sin(nθ)sin(nφ)=cos

(

n(θ−φ)

)

;

d.take the integral outside the series;