1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5. Find the function (x,y) that is harmonic in the upper half-plane Im(z) > 0 and
   has t he boundary values

<fi(x, 0) = 3,

<f>(x, 0) = 1,

for x < -3, and <fi(x, 0) = 7, for - 3 < x < - 1;

for -l<x<2, and (x,0)=4, forx>2.

6. F ind t he function <fi (x, y) that is harmonic in the first quadrant x > O, y > 0 and
   has the boundary values

<f>(O, y) = 0 ,

</>(x, O} = 1,

for y > 1,

for 0 ~ x < l,

and <fi (0, y) = 1, for 0 < y < l ;

and <fi (x, 0) = 0, for x > 1.

7. Find the function (x, y) that is harmonic in the unit disk lz l < 1 and has t he
   boundary values

<fi(x, y) = 0,

<f>(x, y) = 5,

for x + i y = z = eiD, O < 8 < 11';

for x+iy = z = ei^8 , 11' < 8 < 211'.

8. Find the function (x, y) that is harmonic in the unit disk lzl < 1 and has t he
   boundary values

for x + i y = z = ei^9 , 0 < 8 < 11';

forx+iy= z = ei^9 , 11'<8<211'.

<f>(x, y) = 5,

<f>(x, 0) = -5,

for x + iy = z = e;^8 , 0 < 8 < 11';

for-l<x<l.

10. F ind the function </> (x, y) that is harmonic in the portion of the upper half-plane

<f>(x, y) = l ,

<f>(x, 0) = 0 ,

for x + iy = z = e'^9 , 0 < 8 < 11';

for lxl > 1.