Mathematical Analysis

The following problems on Mathematical Analysis can be treated as an assignment for the course MTH275. Please solve the problems (as many as you can) and submit to me.

Real Analysis

Q1: Prove that √2, √3 are irrational numbers.

Q2: State Cauchy’s principle for the convergence of a series. Using this principle show that the series below is not convergent:

$1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots$ $\dfrac{1}{n}+\cdots$.

Q3: Test the convergence of the following series:

1. $\sum \sin \left( \dfrac{1}{n} \right)$
2. $\sum \dfrac{1}{n} \sin \left( \dfrac{1}{n} \right)$
3. $\sum_{n=1}^\infty \dfrac{\sin nx}{n^2} \, (x>0)$
4. $\sum_{n=1}^\infty \dfrac{n}{n^2+\sqrt{n}}$.

Q4: Prove the necessary condition for convergence of a series. Thais is, if the series ∑ u_n is convergent then lim_n→∞ u_n = 0.

Show that the converse is not true by giving an example.

Q5: Show that

$\sum_{n=1}^\infty \dfrac{1}{2n+1}$

is divergent, but the series

$\sum_{n=1}^\infty \dfrac{1}{2n^2+3n}$

is convergent.

Q6: Define an alternating series. Give an example.

Q7: State Leibnitz’s test. Using this test or otherwise, show that the series $1-\dfrac{1}{2^2}+\dfrac{1}{3^2}-$ $\dfrac{1}{4^2}+\cdots$ is convergent.

Q8: Prove that an absolutely convergent series is convergent.

Q9: What is a conditionally convergent series. Prove that the series $1-\dfrac{1}{2}+\dfrac{1}{3}-$ $\dfrac{1}{4}+\cdots$ is conditionally convergent.

Q10: Determine the radius of convergence of the series below:

1. $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3 x^3}{3!}+\cdots$
2. $x+\dfrac{(2!)^2}{4!}x^2+\dfrac{(3!)^2}{6!}x^3+\cdots$ $+\dfrac{(n!)^2}{(2n)!}x^n + \cdots$.

Complex Analysis

Q1: Define the differentiability of a function f(z) at z=z₀. Show that f(z) = |z| is not differentiable at z=0.

Q2: What is an analytic function? Give an example.

Q3: State Cauchy-Riemann equation. Prove that the converse of this statement is not true.

Q4: Find the integral

$\int_{|z|=1} \dfrac{1}{z} \, dz$.

Q5: Evaluate:

$\int_\gamma x \, dz$

where $\gamma$ is the straight line joining 0 to 1+i.

Q6: Using M-L inequality, find the upper bound of

$|\int_\gamma \dfrac{e^z}{z} \, dz|$

where $\gamma$ is the curve $\gamma(t)=e^{it},$ 0 ≤ t ≤ π.

Q7: Using M-L inequality, find the upper bound of

$|\int_\gamma \dfrac{1}{z} \, dz|$

where $\gamma$ is the directed line segment from 1+i to 1+2i.

Q8: State Cauchy integral theorem and formula. Using this or otherwise, evaluate the integral $\int_C \dfrac{z^2-z+1}{z-1} \, dz$ where C is the circle

1. |z| = 1
2. |z| = 1/2.

Q9: Using Cauchy’s integral formula, evaluate the integral

$\int_{|z|=2} \dfrac{e^{4z}}{(z+1)^5} \, dz$.

Q10: Show that the function u(x, y) = 4xy – x³ +3xy² is an harmonic function. Find a function v(x, y) such that f(z) = u+iv is analytic. Express f(z) in terms of z.

Q11: Find the Laurent series expansion of $f(z)=\dfrac{2}{z-3}$ in 0<|z|<1.

Q12: Using residue theorem, find the integral

$\int_{|z|=2} \tan z \, dz$.

Q13: Find the residue of $f(z) =\sin \left( \dfrac{1}{z} \right)$ at the pole $z=0$.

Q14: Find the order of the poles of the function

$f(z) = \dfrac{1}{(z^3-1)(z+1)^2}$.

Also, find the residue at each pole.

Along with the above problems, also study the following:

Must Read:

1. Necessary condition for the convergence of a series with applications.
2. How to find the sum of a geometric series with examples.
3. Last day’s example of Cauchy’s residue theorem, see the shared image in the WhatsApp group.
4. Harmonic conjugate function.