Exercises

Question 1

2.2
Let f : R -> R be uniformly continuous. Show that there exists a, b > 0 such that |f(x)| <= a|x| + b for every x \in R.

Question 2

2.3
Example of f_n -> f differentiable, but f’_n -/-> f’

Question 3

2.4
Example f_n -> f cts. but not diff.

Question 4

3.1
f_n(x) = x/(1+nx^2), determine pointwise and uniform convergence of f_n and f’_n

Question 5

3.2
f_n cts with f_n ->-> f. for every convergent sequence x_n -> x, we have f_n(x_n) -> f(x).
counter example if we only assume pointwise.

Question 6

3.3a
study uniform and pointwise convergence of f_n(x) = x^2/(n^2 + x^2) on R and [0,1].

Question 7

3.3b
Study uniform and pointwise convergence of g_n(x) = x^n/(n+x^n)

Question 8

3.4(a)
if f_n -> on compact K, is it true that f_n ->-> f on K.

Question 9

3.4(e)
If f_n -> f on interval, and f_n increasing, is it true that f also increasing.

Question 10

4.2(a)
Show that g(x) = \sum_{n=1}^{\infty} \cos{2^{n}x}/2^n is cts on R.

Question 11

4.3(b)
If h(x) = \sum_{n=1}^{\infty} 1/(x^{2} + n^{2}), show that h’ is cts.

Question 12

4.4
f_n ->-> f and g_n ->-> g on A. does f_n x g_n ->-> fg on A?

Question 13

4.5
f_n cts. diff. on [a,b], f’->->. show that if there exists x_0 \in [a,b] st f_n(x_0) converges, then f_n ->->

Question 14

4.6
f_n \in C^1([a,b]) ->-> f, is f diff.

Question 15

5.3(a)
f:U -> R^k cts at p. Prove that if f(p) \neq 0, then there exists d > 0 such that |f(x)| > \frac{1}{2} |f(p)| for every x \in U s.t. |x-p| < d.

(b)
suppose that k = 1 and that f(x) \neq 0 for every x \in U. prove that 1/f is cts at p.

Question 16

5.4(b)
g(x,y) = x^3y/(x^2 + y^2)^2. prove that lim_(x,y)->(0,0) g(x,y) does not exist.

Question 17

5.5
f(x,y) = sin(x-y)/(x-y), f(t,t) = 1, prove that f is cts on R.

Question 18

6.4
Prove that U = {(a,b,c,d) \in R^4 : |ad - bc| > 1} is open.

Question 19

7.1(a)
A \in R^n,n , B_j = \sum_{r=0}^{j} A^r/r!.
Prove that B_j is Cauchy wrt operator norm.

(b)
B_j -> e^A, compute e^A when A = (0 t // -t 0)

Question 20

8.1
for v \in R^n, let Ax = v . x. Prove that ||A||op = ||v||

Question 21

8.2
Show that if A \in GL(n,R) and ||A^-1||op = u, then ||Ax|| >= 1/u||x|| for every x \in R^n

Question 22

8.3
Prove that Df(x) is unique.

Question 23

8.4(d)
verify that f_X(x,y) = 2xy(y^2 - 2x^6)/(x^6 + y^2)^2 is unbounded on B(0, d) for every d>0.

Question 24

8.5
f:R^n -> R, r:(ab) -> R^n cts. diff. and r solves r’(t) = -\nabla f(r(t)). Prove that if [\alpha, beta] \subset (a,b), then f(r(\alpha)) >= f(r(\beta)), with equality if and only if \nablda f(r(\alpha)) = 0 and r()t) = r(\alpha) for every t \in [\alpha,. \beta].

Question 25

9.1
If f(x _ iy) = Q(x) + i P(y) is analytic, determine the most general form of Q and P.

Question 26

9.4
Find the harmonic conjugate for u(x,y) = sinx coshy

Question 27

10.1

(a) \int_|z-1|=1 \bar{z} dz

(b) \int_|z-|=1 \bar{z} |dz|

(c) \int_|z|=2 |z+1|^2 dz

(d) \int_|z|=2 (x+1)^2 |dz|

Question 28

10.2
Integrate f(z) = z^2 over a wedge of 45degrees and radius 1.

Question 29

10.3
Let f be cts on C st lim(z->\infty) zf(z) = k. Let D be the curve z = Re^{it} with 0 <= t <= \theta ACW.

(a)
show that \lim_{R->\infty} \int_{D} f(z) dz = ik\theta

Question 30

10.4
If \Omega \subset C open and f:\Omega -> C is cts, prove that |\int_{\gamma} f(z) dz| \leq \max{z \in \gamma} |f(z)|

Question 31

10.5
Compute \int_{B_{R}(0} z^{n} d\bar{z}

Question 32

10.6
If f:C-> C and |f| are analytic, show that f must be constant.

Question 33

11.1
f analytic on \Omega \subset C and a \in \Omega with B_{r}(a) \subset \Omega. Show that f(a) = (1/2\pi) \int_{0}^{2\pi} f(a + re^{i\theta}) d\theta

Question 34

11.3
f:C -> C analytic st |f(z)| <= K|z|^{k} for K, k > 0.
Prove that f is a polynomial of degree at most k.

Question 35

11.4
Evaluate \int_{0}^{\infty} 1/(1+x^{3}) = \lim_{R -> \infty} 1/(1+x^{3}) using wedge of 2\pi/3 of radius R and circle around positive third root of unity.

Question 36

7.3.3 Example 1
Compute \int_{-\infty}^{\infty} 1/(1+x^{2}) dx using a arc and circle around the singulatity.

Question 37

7.3.3 Example 2
Compute \int_{-\infty}^{\infty} 1/(1+x^{4})

Question 38

7.3.3 Example 3
Compute \int_{-\infty}^{\infty} cos3x/(4+x^{2})

Question 39

7.3.3 Example 4
Compute \int_{-\infty}^{\infty} xsinx/(1+x^{2})

Question 40

Prove Louivilles Theorem

Question 41

Prove Weierstrass M-test