Question

Question 1 (15 marks) Let f be a holomorphic, non-constant function on a domain 2 c C, and choose any open set UC 2. Define a

0 0
Add a comment Improve this question Transcribed image text
Answer #1

holomorphic & Non constant we 20 ਆਗਰਮ U is open set avis image of u under f het wotf(u). Then there exist 7 EU f(zo)- Wo is o

Add a comment
Know the answer?
Add Answer to:
Question 1 (15 marks) Let f be a holomorphic, non-constant function on a domain 2 c...
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for? Ask your own homework help question. Our experts will answer your question WITHIN MINUTES for Free.
Similar Homework Help Questions
  • 4. Let p(u, v) be a non-zero Cl function of two real variables whose gradient is non-zero on the ...

    4. Let p(u, v) be a non-zero Cl function of two real variables whose gradient is non-zero on the set fp 0, and let f u+ iv be holomorphic on region 2 C C and satisfy p(Re (f), Im (f))-0. Prove that f is constant on Ω. Conclude as special cases that if f is holomorphic on a connected open set and f is real valued, then f is constant, or if the modulus off is constant on Ω, then...

  • 1. Let k E C and define f:C+C to be the constant function f(x) = k....

    1. Let k E C and define f:C+C to be the constant function f(x) = k. Use part (1) or (2) of to prove that f is continuous. (1) For every closed set FCC, the preimage f-1(F) is F' n D for some closed set F' (2) For every open set G C C, the preimage f-1(G) is G'' n D for some open set G'.

  • Please answer this question Implicit Function Theorem in Two Variables: Let g: R2 - R be...

    Please answer this question Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above (2) Since dg(a, b)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...

  • Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...

    Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...

  • Log(2 - 2) (x2 y Question 2. Consider the function f(x, y, (a) What is the maximal domain of f? (...

    log(2 - 2) (x2 y Question 2. Consider the function f(x, y, (a) What is the maximal domain of f? (Write your answer in set notation.) (b) Find ▽f. (c) Find the tangent hyperplnes Te2)(r, y,z) and Tao2-)f(x, y, z). Find the intersection of these two hyperplanes, and very briefly describe the intersection in words (0,1, 1) and set notation. Confirm that the point (2, 2, 1) is on this level surface, and that Vf(2, 2, 1) is (d) On...

  • Question 3. (4 marks) Let C([a, b]; R) be the space of all continuous functions on...

    Question 3. (4 marks) Let C([a, b]; R) be the space of all continuous functions on [a, b], 0 <a<b with the metric || f – 9|| = maxasaso \f (x) – g(x)]. For each f e C([a, b]; R), define a map F(f) by F(f)(x) = x5 + Vx € (a,b]. (65 – a5) Prove that there is a unique fixed point of F in the space C([a, b]; R); i.e. there is a unique fe C([a,b); R) such...

  • 2. Let V and W be vector spaces over F. Define the set v, w) |v...

    2. Let V and W be vector spaces over F. Define the set v, w) |v V andwEW This is called the product of V and W (a) Show that V x W is a vector space. (b) Define a map w : V → V × W by w (z) = (z,0). Show that w is an injective linear map. Note that we can define a similar map lw (c) If (d) Show that V x W. (V W...

  • surface patch for S. regular surface and f: S Ra smooth EXERCISE 3.44. Let S be...

    surface patch for S. regular surface and f: S Ra smooth EXERCISE 3.44. Let S be a function. Assume that the point p e S is a critical point of f, which means that dfp(v) 0 for all v e TpS. Define the Hessian of f atp in the direction v as Hess(f)p(v) (foy)"(0), where y is a regular curve in S with y(0) = p and y'(0) = v. Prove that the Hessian is well defined in the sense...

  • 1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an exa...

    1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an example of an unbounded open set with finite measure. Justify your answer, 3) If a is a single point on the number line show that m ( a ) = O. 4) Prove that if K is compact and U is open with K U then m(K) m(U). 5) show that the Cantor set C is compact and m(C)...

  • 08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty...

    08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...

ADVERTISEMENT
Free Homework Help App
Download From Google Play
Scan Your Homework
to Get Instant Free Answers
Need Online Homework Help?
Ask a Question
Get Answers For Free
Most questions answered within 3 hours.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT