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Polynomials Let n be an integer. Show that ~n is an entire function both from the...
Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,q) = $' p(x)q(x)"(x) dx. You do not need to check that this is an inner product. (a) Prove that P.(R) has an orthonormal basis po..., Pr such that deg pk = k for each k. (b) Show that (Pk, pk) = 0 for each k, where the polynomials pį are from the preceding part. Here pé denotes...
l. Assume that j : R-→ R-s C and satisfies what are known as the Cauchy-Riemann equations: (c) Let r-(r1, 2) and (s1, s2) be vectors in IR2 and suppose that (ri, 2)f(s1, 82) and Df(81,82)メ0. Show that f-1 satisfies the Cauchy-Riemann equations when evaluated at r. (Hint: Might I make a notational suggestion: Leta(s) = sim) = % (n, s) and b(s) 쓺(81, 82) =-警( )) 81,82 (d) For this last bit, drop the assumption that f satisfies the...
11. Show that following are an entire function. f(z)= e-? f(3) = cosh(z) (a) (b) 12. Find the derivative of the following using Cauchy- Reimann equations: ali d [2z - i dz z + 2i Z + 2z3 (a) (b)
3. Let f be an entire function whose modulus is contant on a circle centred at a. Show that f(z) = c(z - a)" for some integer n > 0 and a constant ceC.
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Let {dn}n≥0 denote the number of integer solutions a1 +a2 +a3 +a4 = n where 0 ≤ ai ≤ 5 for each i = 1, 2, 3, 4. Write the ordinary generating function for {cn}n≥0. Please express the ordinary generating function as a rational function p(x) /q(x) where both p(x) and q(x) are polynomials in the variable x.
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5 Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5
Let n be a positive integer with n > 20 , and let with 1. Show that S possess two disjoint subsets, the sum of whose elements are equal. S 1,2,., 1n2) We were unable to transcribe this image
(b) Let D C C be a regular domain, f : D → D' C C be a complex-valued function and f(z) = u(x,y) + iv(x,y). (a) Show that if/(z) is differentiable on D implies the Cauchy-Riemann equation, i.e., au dyJu on D. (b) Assume that D- f(D).e. fis a conformal mapping from domain D onto domain D. Le x' =a(x,y), y = r(x,y). Show that if φ(x,y) is harmonic on D. ie..知+Oy-0, then is also harmonic on domain D....
Let A E(R") be Hermitian and positive definite, let v Define g R" R by R", and let cE R (a) Show that g is polynomial function of (... ,En) and in particular it has continuous partial derivatives of all orders. (b) Show that oo. Hint: Use Ezercise Ic. (c) Prove that g(x) achieves a global minimum d) Compute Vg(x). Show that g has a unique critical point, and hence argue that the minimum must be achieved at this point....