Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Big O notation wikipedia , lookup
History of the function concept wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Line (geometry) wikipedia , lookup
Factorization wikipedia , lookup
Vincent's theorem wikipedia , lookup
Non-standard calculus wikipedia , lookup
Fundamental theorem of calculus wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Math 611 Assignment # 4 1. Suppose C is a boundary of a simply-connected domain D ⊂ C oriented in the counterclockwise direction. Recall the argument principle: If f (z) is meromorphic inside D and is analytic and not equal to zero at all points on the boundary C. Then I 0 1 1 f (z) ∆C arg(f ) = dz = N0 − Np , 2π 2πi C f (z) where N0 is the number of zeros (w.r.t. algebraic multiplicity) inside D and Np is the number of poles (w.r.t. pole orders) inside D. Now, consider ξ ∈ C and a function f (z) meromorphic inside D and analytic on the boundary C. Suppose f (z) 6= ξ and f (z) 6= 0 for all points z on the boundary C. Suppose also f (z) 6= 0 for all z ∈ D (i.e. N0 = 0). Explain the meaning of I I 0 f 0 (z) f (z) 1 1 dz − dz 2πi C f (z) − ξ 2πi C f (z) 2. How many roots of ez − 3z 3 are there inside the unit circle B1 (0) around the origin? Hint: Find f and g such that f (z) + g(z) = ez − 3z 3 and |f (z)| > |g(z)| for all z in {z : |z| = 1}. Use Rouché’s Theorem. 3. How many zeros does the polynomial z 7 + 4z 4 + z 3 + 1 have in the regions {|z| < 1} and {1 < |z| < 2}? Hint: Compare |z 7 + 4z 4 | to |z 3 + 1| on {|z| = 1}, and use Rouché’s Theorem. 4. Show that all five roots of the algebraic equation z 5 + 15z + 1 must be situated in the interior of the circle |z| < 2, but that only one root of this equation is in the circle |z| < 32 . 5. Show that the equation zea−z = 1, where a ∈ R and a > 1, has precisely one root in the circle |z| ≤ 1. 6. Show that, in |z| < 1, the function f (z) = z + takes every nonreal value exactly once. 1 z 7. Let f (z) be analytic in a domain D and on its boundary C. If |f (z)| = 1 for all z ∈ C, show that unless f (z) is a constant there must be at least one zero of f (z) in D. 1 Hint: Consider f (z) . 8. (1998 qual, # 2) Let f (z) be analytic function in the entire complex plane. Suppose there exist real numbers M and α ≥ 0 such that |f (z)| ≤ M (1 + |z|)α . Prove that f (z) is a polynomial of degree less than or equal to α. 9. (1998 qual, # 3) Let f (z) and g(z) be analytic functions in the open disc |z| < 2. Assume (a) |f (z)| ≥ |g(z)| for any z with |z| = 1 and (b) f (z) is not zero for any z with |z| < 1. Prove that |f (z)| ≥ |g(z)| for any z with |z| < 1. Give an example which shows that this conclusion is not true without assumption (b). 10. (2005 qual, # 3) (a) Suppose a function f is analytic everywhere in the complex plane except for a finite number of singular points interior to a circle Γ, |z| = r, r > 0, which is traversed once in the counterclockwise direction. Suppose Res{f (z), z0 } denotes the residue of f (z) at z0 . Prove the equality I 1 1 f (z)dz = 2πi Res f , 0 . 2 z z Γ (b) Suppose P (z) and Q(z) are two complex polynomials of degrees n and m respectively. Suppose Γ is a simple closed contour that encloses all zeros of Q(z). If m ≥ n + 2 show that the contour integral I P (z) dz = 0 . Q(z) Hint: do a change of variable w = z1 .