View the step-by-step solution to:

University of Toronto - Mississauga MAT334S - Winter 2016 Assignment 2 - due Friday, January 29th, at the beginning of your tutorial Problem 1....

asking mat334h5 assignment solution

a2-16.dvi

Problem 1. Compute lim z2−2i , if it exists. (Caution: if you make the direct substitution z→1+i z2−2z+2

z = 1 + i, you will get 0/0 ...)
Problem 2. Compute limz→0

University of Toronto – Mississauga MAT334S – Winter 2016 Assignment 2 - due Friday, January 29th, at the beginning of your tutorial Problem 1. Compute lim z 1+ i z 2 2 i z 2 2 z +2 , if it exists. (Caution: if you make the direct substitution z = 1 + i , you will get 0 / 0 ...) Problem 2. Compute lim z 0 ( z ¯ z + ¯ z z ) , if it exists. (Of course, this time it does not: use the exponential form z = rexp ( ) of z to verify this!) Problem 3. Here we shall compute derivatives the long way (for a simple function), and the short way (for a complicated function). Compute f ( z 0 ) for f ( z ) = z 2 by using the deFnition directly (i.e. by computing the limit lim Δ z 0 f ( z 0 z ) f ( z 0 ) Δ z ). Use the rules of di±erentiation and the chain rule to compute f ( z ) for f ( z ) = ze z 2 . Here, the exponent of z = x + iy is deFned by Euler’s formula: exp ( x + iy ) = exp ( x ) · exp ( iy ) = e x (cos y + i sin y ). Problem 4. Show that f ( z ) = ¯ z is nowhere di±erentiable (i.e. there is no point z 0 C such that f ( z 0 ) exists). Problem 5. Check that f ( z ) = e z + z + 1 is an entire function by using the Cauchy-Riemann equations and su²cient conditions for di±erentiability. Problem 6. Let f ( z ) be a di±erentiable function. Suppose that Re f ( z ) = x 2 - y 2 + y if we substitute z = x + iy into f ( z ). By using the Cauchy-Riemann equations, reconstruct Im f ( z ), up to an additive constant c C . Problem 7. Let f ( re ) = u ( r, θ ) + iv ( r, θ ) be given in polar coordinates by u ( r, θ ) = r ( r 2 - 1) cos θ and v ( r, θ ) = r ( r 2 + 1) sin θ . ³ind all points in C at which the Cauchy-Riemann equations are satisFed for f ( z ). (Since here we put z = re , use the polar form of Cauchy- Riemann equations) 1
Background image of page 1
Sign up to view the entire interaction

Top Answer

Detailed answer. feel... View the full answer

cr1.jpg

Sign up to view the full answer

Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.

-

Educational Resources
  • -

    Study Documents

    Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

    Browse Documents
  • -

    Question & Answers

    Get one-on-one homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!

    Ask a Question
Ask a homework question - tutors are online