IAS PYQs 2
1994
1) Suppose that \(z\) is the position vector of a particle moving on the ellipse \(C :z = a \cos \omega t + i b \sin \omega t\) where \(a, b, \omega\) are positive constants, \(a>b\) and \(t\) is the time. Determine where
(i) the velocity has the greatest magnitude.
(ii) the acceleration has the least magnitude
[10M]
2) How many zeros does the polynomial \(p ( z )= z ^{4}+2 z ^{3}+3 z +4\) possess in (i) the first quadrant, (ii) the fourth quadrant.
[10M]
3) Test of uniform convergence in the region \(\vert z\vert \leq 1\) the series \(\sum_{n=1}^{\infty} \dfrac{\cos n z}{n^{3}}\)
[10M]
4) Find Laurent series for (i) \(\dfrac{e^{2 z}}{(z-1)^{3}}\) about \(z=1\), (ii) \(\dfrac{1}{z^{2}(z-3)^{2}}\) about \(z=3\).
[10M]
5) Find the residues of \(f(z)=e^{z} \operatorname{cosec}^{2} z\) at all its poles in the finite plane.
[10M]
6) By means of contour integration, evaluate \(\int_{0}^{\infty} \dfrac{\left(\log _{e} u\right)^{2}}{u^{2}+1} d u\)
[10M]
1993
1) In the finite \(z\) plane, show that the function \(f(z)=\sec \dfrac{1}{z}\) has infinitely many isolated singularities in a finite intervals which includes \(0 .\)
[10M]
2) Find the orthogonal trajectories of the family of curves in the xy-plane defined by \(e-x(x \sin y-y \cos y)=\alpha\) where \(\alpha\) is real constant.
[10M]
3) Prove that (by applying Cauchy Integral formula or otherwise) \(\int_{0}^{2 \pi} \cos ^{2 n} \theta d \theta=\dfrac{1,3,5 \ldots(2 n-1)}{2,4,6 \cdot 2 n} 2 \pi\) where \(n =1,2,3...\)
[10M]
4) If \(c\) is the curve \(y = x ^{3}-3 x ^{2}+4 x -1\) joining the points (1,1) and (2,3) find the value of \(\int_{c}\left(12 z^{2}-4 i z\right) d z\)
[10M]
5) Prove that \(\sum_{n=1}^{\infty} \dfrac{z^{n}}{n(n+1)}\) converges absolutely for \(\vert z\vert \leq 1\)
[10M]
6) Evaluate \(\int_{0}^{\infty} \dfrac{d x}{x^{6}+1}\) by choosing an appropriate contour.
[10M]
1992
1) If \(u=e^{-x}(x siny -y \cos y),\) find \(v\) such that \(f(z)=u+i v\) is analytic. Also find \(f(z)\) explicitly as a function of \(z\).
[10M]
2) Let \(f(z)\) be analy tic inside and on the circle \(C\) defined by \(\vert z\vert =R\) and let \(z=e r^{i\theta}\) be any point in side C. Prove that \(f\left(r e^{i \theta}\right)=1 / 2 \pi \int_{0}^{2 \pi} \dfrac{\left.\left(R^{2}-r^{2}\right) \left(Re^{i\phi}\right)\right)}{R^{2}-2 R r \cos (\theta+\phi)+r^{2}} d \phi\)
[10M]
3) Prove that all the roots of \(z^{7}-5 z^{3}+12=0\) lie between the circle \(\vert z\vert =1\) and \(\vert z\vert =2\).
[10M]
4) Find the region of convergence of the series whose \(n\)-th term is \(\dfrac{(-1)^{n-1} z^{2 n-1}}{(2 n-1) !}\)
[10M]
5) Expand \(f(z)=\dfrac{1}{(z+1)(z+3)}\) in a Laurent series valid for 6.b)(i) \(\vert z\vert >3\) 6.b)(ii) \(1<\vert z\vert <3\) 6.b)(iii) \(\vert z\vert <1\)
[10M]
6) By integrating along a suitable contour evaluate \(\int_{0}^{8} \dfrac{\cos m x}{x^{2}+1} d x\)
[10M]
1991
1) A function \(f(z)\) is defined for finite values of \(z\) by \(f(0)=0\) and \(f(z)=e^{-z^{-4}}\) everywhere else. Show that the Cauchy Riemann equation are satisfied at the origin. Show also that \(f(z)\) is not analytic at the origin.
2) If \(\vert \mathrm{a}\vert \neq \mathrm{R}\) show that \(\int_{\vert z \vert =R} \dfrac{\vert dz \vert}{\vert z-a\vert \vert z+a \vert}< \dfrac{2 \pi R}{\vert R^{2}-\vert a \vert^{2} \vert}\).
3) If \(J_{n}(t)=\dfrac{1}{2 \pi} \int_{0}^{2 z} \cos (n \theta-t \sin \theta) d \theta\). Show that \(e^{\dfrac{1}{2}\left(z-\dfrac{1}{z}\right)}\)=\(J_{0}(t)+z J_{1}(t)+z^{2} J_{2}(t)\)+ \(\cdots\)-\(\dfrac{1}{z} J_{1}(t)+\dfrac{1}{z^{2}} J_{2}(t)-\dfrac{1}{z^{3}} J_{3}(t)+ \cdots\)
4) Examine the nature of the singularity of \(e^{z}\) at infinity.
5) Evaluate the residues of the function \(\dfrac{Z^{3}}{(Z-2)(Z-3)(Z-5)}\) at all singularities and show that their sum is zero.
6) By integrating along a suitable contour, show that \(\int_{-\infty}^{\infty} \dfrac{e^{a x}}{1+e^{x}}=\dfrac{\pi}{\sin a \pi} \quad\) where \(0<\mathrm{a}<1\).
1990
1) Let \(f\) be regular for \(\vert Z\vert <R\), prove that, if \(0<r<R\), \(f^{\prime}(0)=\dfrac{1}{\pi r} \int_{0}^{2 \pi} u(\theta) e^{-i d} d \theta\), where \(u(\theta)=\operatorname{Re} f\left(r e^{i \theta}\right)\).
2) Prove that the distance from the origin to the nearest zero of \(f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}\) is at least \(\dfrac{r\vert a_{0}\vert }{M+\vert a_{0}\vert } \cdot\) where \(\mathrm{r}\) is any number not exceeding the radius of the convergence of the series and \(M=M(r)=\sup \vert f(z)\vert _{\vert z\vert =r}\).
3) Prove that \(\int_{-\infty}^{\infty} \dfrac{x^{4}}{1+x^{8}} d x=\dfrac{\pi}{\sqrt{2}} \sin \dfrac{\pi}{8}\) using residue calculus.
4) Prove that if \(\mathrm{f}=\mathrm{u}+\mathrm{iv}\) is regular through out the complex plane and \(au+\) bv-c \(\geq 0\) for suitable constants \(a\), \(b\), \(c\) then \(\mathrm{f}\) is constant.
5) Derive a series expansion of \(\log \left(1+e^{z}\right)\) in powers of \(z\).
6) Determine the nature of singular points \(\sin \left(\dfrac{1}{\cos 1 / z}\right)\) and investigate its behaviour at \(z=\infty\).