Show that u(x, y)= 2x –x' +3xy´ is harmonic and find a harmonic conjugate v(x, y). Show that exp(z²) s exp(z[*) for all z e C. 5. 6. Show that Log[(-1+i)’]# 2Log(-1+i). 7. Find all roots of the...

Question 9 Question 10

Extracted text: Show that u(x, y)= 2x –x' +3xy´ is harmonic and find a harmonic conjugate v(x, y). Show that exp(z²) s exp(z[*) for all z e C. 5. 6. Show that Log[(-1+i)’]# 2Log(-1+i). 7. Find all roots of the equation log(z) = 7i / 2 8. Find the principal value of (1+ i)'. 9. Use the definitions of sin(z) and cos(z) given in Lecture 13 at the 13:30 mark to prove that sin z+ = cos(z) for all z E C. (Do not use any other trigonometry identities for question 9) 10. Evaluate(3t - i)²dt