Question #1
Circle #1 is centered at 1 + i8 and has radius 1. Circle #2 is centered at (1+i8+(3(1)+2i) and has radius 1(1 + 1/6). Circle #1 has temperature 0 degrees, and circle #2 has temperature 1 degree.
What is the temperature at z = (1-2(1))-i6, accurate to ±0.01 ?
Question #2
Circle #1 is the same as in Problem 1. A straight line is tangent to the circle at (1+i8)+(1)(1+i)/v2; so it has slope -1. The circle has temperature 00 and the straight line has temperature 10.
What is the temperature at z = ((1)-2(1))-i6, accurate to ±0.01 ?
Question #3
Circle #1 is the same as in Problem 1. Circle #3 is centered at (1+1)+i8 and has radius 1[1 + 1/(6+1)]. Circle #1 has temperature 00 and circle #3 has temperature 10. What is the temperature at z = (1+1/2) + i8 + i1/10, accurate to ±0.01 ?
Advice
Use software to help you with each of these. Maintain full computer accuracy until the final answer, and then round.
First do any preliminary mapping that may be necessary to bring the geometry into “standard position and dimensions,” if necessary. Then formulate a mapping (there may be many) that simplifies the region. Since you have software, you can quickly
evaluate your map at 5 points along each of the boundaries, to ensure that you know exactly what the transformed geometry looks like. Also evaluate it at the requested point
z, to see which subregion is relevant.
Then label the temperatures in the transformed subregion, and formulate the solution to Laplace’s equation there.
Then insert the z value into the (possibly composite) mapping to get the transformed point, and evaluate the temperature.
As a check, also evaluate the temperature at several z-points on the boundaries, where its value is known.