Homework Answers
![We have ft C = } rett , ato 62 a po , a O eni na o Now a etx +t(n) = a aeta ea, e -/ -(1-) = neta - [+ ++ + -] [1-(+*+-)+(462](http://img.homeworklib.com/questions/688f1030-c9ef-11ea-90cc-35112993de71.png?x-oss-process=image/resize,w_560)
4. Define the function f(x) = x/(e* – 1) so that f(0) = 1, then the...
4. Define the function f(x) = x/(e* – 1) so that f(0) = 1, then the Maclaurin expansion of f(x) has a positive radius of convergence. a) When the Maclaurin expansion is expressed in the form B2 B3 3 i=1+ B13 +31 x2 + x3 + ... + x" + ... n! the coefficients B1, B2, B3,... are called the Bernoulli numbers. Substitute the Maclaurin series for et and use polynomial long-division to find the first four Bernoulli numbers. b)...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and comple...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We d...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
8. We define the polynomials To(x), T1(x), Ta(x), . . . by the following properties: (a)...
8. We define the polynomials To(x), T1(x), Ta(x), . . . by the following properties: (a) they are orthogonal with respect to the weight function p= (1-)on the interval -1 < x < 1; (b) Ta is of degree n; (c) Tn(1) 1/Show that this determines.the Ta, and verify that the functions T.(x) COS (n COs-x) satisfy the conditions. (The Tn are called Tchebysheff polynomials.)
8. We define the polynomials To(x), T1(x), Ta(x), . . . by the following properties:...
5. For t ER, define the evaluation map evt : Pn(R) + R given by evt(p(x))...
5. For t ER, define the evaluation map evt : Pn(R) + R given by evt(p(x)) = p(t). Here we consider R as the vector space R1. (a) Prove evt is a linear map. (b) For part (b), let n= 4. Write down a polynomial p e ker(ev3). (c) For any t, the set of polynomials Ut = {p E Pn(R) : p(t) = 0} is a subspace. What is the dimension of Ut (in terms of n)? Justify your...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials Co, ..., (n of degree n by (r - k) Let P, = 1,[r] be the polynomials of degree n, which is a vector space of dimension n + 1. (a) Show that the n+1 polynomials {lo, ..., Ln^ are basis for P i.e., they are linearly independent. (b) Find the coordinates [f]в of polynomial f E 1, with respect to the basis l-[10,...
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10...
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for each n x n matrix A, we define the n x n matrix p(A) by P(A) = ao In + a A+ ... + amA”. Also, for each n, let Onxn E Rnxn be the n x n zero matrix. (a) Show that for all polynomials p and q and square matrices A, we have p(A)q(A) = 9(A)p(A). (b) Show that for every 2...
(3) The Hypergeometric Function If a, b, c R with c f {0, -1,-2,...^ we define the Gauss hypergeo...
I just need 3d answered please!
(3) The Hypergeometric Function If a, b, c R with c f {0, -1,-2,...^ we define the Gauss hypergeometric function as n!c(c 1)... (c+n-1) Note that this solves the DE (a) Verify that log(1x) rF(1,1,2, -) (b) Verify formally (without justifying the limits) that e-lim F (a, b, a, (c) Show that Pla, b, c, x) = abF(a + 1,D+ 1, c + 1, x) (d) Show that F(n, -n, s a polynomial, and...
q1 1. Consider the alphabet set Σ = (0,1,2) and the enumeration ordering on Σ*, what...
q1
1. Consider the alphabet set Σ = (0,1,2) and the enumeration ordering on Σ*, what are the 20th and 25th elements in this ordering? 2. Let N be the set of all natural numbers. Let S1 = { Ag N is infinite }, S2-( A N I A is finite) and S-S1 x S2 For (A1,B1) E S and (A2,B2) E S, define a relation R such that (A1,B1) R (A2,B2) iff A1CA2 and B2CB1. i) Is R a...
4. Define the function f(x) = x/(e* – 1) so that f(0) = 1, then the Maclaurin expansion of f(x) has a positive radius of convergence. a) When the Maclaurin expansion is expressed in the form B2 B3 3 i=1+ B13 +31 x2 + x3 + ... + x" + ... n! the coefficients B1, B2, B3,... are called the Bernoulli numbers. Substitute the Maclaurin series for et and use polynomial long-division to find the first four Bernoulli numbers. b)...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
8. We define the polynomials To(x), T1(x), Ta(x), . . . by the following properties: (a) they are orthogonal with respect to the weight function p= (1-)on the interval -1 < x < 1; (b) Ta is of degree n; (c) Tn(1) 1/Show that this determines.the Ta, and verify that the functions T.(x) COS (n COs-x) satisfy the conditions. (The Tn are called Tchebysheff polynomials.)
8. We define the polynomials To(x), T1(x), Ta(x), . . . by the following properties:...
5. For t ER, define the evaluation map evt : Pn(R) + R given by evt(p(x)) = p(t). Here we consider R as the vector space R1. (a) Prove evt is a linear map. (b) For part (b), let n= 4. Write down a polynomial p e ker(ev3). (c) For any t, the set of polynomials Ut = {p E Pn(R) : p(t) = 0} is a subspace. What is the dimension of Ut (in terms of n)? Justify your...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials Co, ..., (n of degree n by (r - k) Let P, = 1,[r] be the polynomials of degree n, which is a vector space of dimension n + 1. (a) Show that the n+1 polynomials {lo, ..., Ln^ are basis for P i.e., they are linearly independent. (b) Find the coordinates [f]в of polynomial f E 1, with respect to the basis l-[10,...
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for each n x n matrix A, we define the n x n matrix p(A) by P(A) = ao In + a A+ ... + amA”. Also, for each n, let Onxn E Rnxn be the n x n zero matrix. (a) Show that for all polynomials p and q and square matrices A, we have p(A)q(A) = 9(A)p(A). (b) Show that for every 2...
I just need 3d answered please!
(3) The Hypergeometric Function If a, b, c R with c f {0, -1,-2,...^ we define the Gauss hypergeometric function as n!c(c 1)... (c+n-1) Note that this solves the DE (a) Verify that log(1x) rF(1,1,2, -) (b) Verify formally (without justifying the limits) that e-lim F (a, b, a, (c) Show that Pla, b, c, x) = abF(a + 1,D+ 1, c + 1, x) (d) Show that F(n, -n, s a polynomial, and...
q1
1. Consider the alphabet set Σ = (0,1,2) and the enumeration ordering on Σ*, what are the 20th and 25th elements in this ordering? 2. Let N be the set of all natural numbers. Let S1 = { Ag N is infinite }, S2-( A N I A is finite) and S-S1 x S2 For (A1,B1) E S and (A2,B2) E S, define a relation R such that (A1,B1) R (A2,B2) iff A1CA2 and B2CB1. i) Is R a...
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