Engineering Mathematics - II

If $ f(z)=u(x,y)+iv(x,y) $ is analytic function, then $ f'(z)= $

For an analytic function $ f(z)=u(x,y)+i v(x,y) $, if the real part $ u(x,y)=x^{2}-y^{2} $ is given, what is the imaginary part $ v(x,y) $?

The value of the integral $ \oint_C |z|dz $, where C is the left half of the unit circle $ |z|=1 $ from $ z=-i $ to $ z=i $

The residue of $ e^{z}/\cos \pi z $ at its pole $ z=-1/2 $ is

The particular integral $ y_p $ of the differential equation $ (D^{2}+4)y=\cos 2x $, $ D \equiv \frac{d}{dx} $ is

By the method of undetermined coefficients, the particular integral $ y_p $ of the differential equation $ (D^{4}+2D^{3}+D^{2}) y=12x^{2} $, $ D \equiv \frac{d}{dy} $ is

Let $ \lim_{n\rightarrow\infty}(u_{n})^{1/n}=l $, then the series $ \sum_{n=1}^{\infty}u_{n} $ converges if

$ L(\sin at)=\frac{a}{s^{2}+a^{2}} $ when

If $ L^{-1}(\frac{3s+8}{s^{2}+4s+25})=e^{kt}[3 \cos\sqrt{21}t+\frac{2}{\sqrt{21}}\sin\sqrt{21}t] $ then the value of k is

If $ f(x)=\sqrt{\frac{1-\cos x}{2}}, 0 < x < 2\pi $ then the value of $ a_0 $ is

Find $ \lim_{z\rightarrow 0} [\frac{1}{1-e^{1/x}} + iy^2] $ if it exists.

Check the continuity of the following function at $ z=0 $:
$ f(z)=\begin{cases} \frac{Re(z^{2})}{|z|^{2}}, & z \ne 0 \\ 0, & z=0 \end{cases} $

Examine whether the Cauchy-Riemann equations are satisfied at the origin for the function given below or not:
$ f(z)=\begin{cases} \frac{x^{3}(1+i)-y^{3}(1-i)}{x^{2}+y^{2}}, & z \ne 0 \\ 0, & z=0 \end{cases} $
Also check, whether $ f(z) $ is analytic at $ z=0 $ or not.

If $ f(\xi)=\int_{C}\frac{3z^{2}+7z+1}{z-\xi}dz, $ where C is an ellipse $ 9x^{2}+16y^{2}=144 $. Find the values of $ f(3) $, $ f'(1-i) $ and $ f''(1-i) $.

Evaluate the integral $ \int_{0}^{\infty}\frac{\sin ax}{x(x^{2}+b^{2})}dx $, $ a>0, b>0 $ by the use of Cauchy residue theorem.

Find the Laurent's expansion for $ f(z)=\frac{7z-2}{z^{3}-z^{2}-2z} $ in the region $ 1 <|z+1|<3 $.

Find the complete solution of $ \frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=x+e^{x}\cos x $.

Find the solution of $ x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+y=\log x \cdot \sin(\log x) $.

Find the solution of $ \frac{d^{2}y}{dx^{2}}-6\frac{dy}{dx}+9y=\frac{e^{3x}}{x^{2}} $ by the method of variation of parameters.

Test the convergence of the series $ \Sigma(1+\frac{1}{n})^{-n^2} $.

Test the convergence of the series $ \frac{a}{b}+\frac{a+x}{b+x}+\frac{a+2x}{b+2x}+\frac{a+3x}{b+3x}+...... $

Find Laplace transform of $ t \cos at $.

Evaluate the integral $ \int_{0}^{\infty}te^{-3t}\sin t dt $, by the use of Laplace transform.

Find the inverse Laplace transform of $ \tan^{-1}(\frac{2}{s^{2}}) $.

Find the solution of the following initial value problem by the use of Laplace transform:
$ \frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}-y=0, $ if $ y(0)=0 $ and $ (\frac{dy}{dt})_{t=0}=1. $

Find the Fourier series expansion of the following function:
$ f(x)=\begin{cases} 0, & -\pi \le x \le 0 \\ \sin x, & 0 \le x < \pi \end{cases} $
Hence deduce that $ \frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{5.7}-\frac{1}{7.9}+......=\frac{\pi-2}{4} $.

Find the Fourier expansion of $ x\sin x $ as Fourier cosine series in the interval $ [0,\pi] $.

If $ f(z)=u(x,y)+iv(x,y) $ is analytic function, then $ f'(z)= $

For an analytic function $ f(z)=u(x,y)+i v(x,y) $, if the real part $ u(x,y)=x^{2}-y^{2} $ is given, what is the imaginary part $ v(x,y) $?

The value of the integral $ \oint_C |z|dz $, where C is the left half of the unit circle $ |z|=1 $ from $ z=-i $ to $ z=i $

The residue of $ e^{z}/\cos \pi z $ at its pole $ z=-1/2 $ is

The particular integral $ y_p $ of the differential equation $ (D^{2}+4)y=\cos 2x $, $ D \equiv \frac{d}{dx} $ is

By the method of undetermined coefficients, the particular integral $ y_p $ of the differential equation $ (D^{4}+2D^{3}+D^{2}) y=12x^{2} $, $ D \equiv \frac{d}{dy} $ is

Let $ \lim_{n\rightarrow\infty}(u_{n})^{1/n}=l $, then the series $ \sum_{n=1}^{\infty}u_{n} $ converges if

$ L(\sin at)=\frac{a}{s^{2}+a^{2}} $ when

If $ L^{-1}(\frac{3s+8}{s^{2}+4s+25})=e^{kt}[3 \cos\sqrt{21}t+\frac{2}{\sqrt{21}}\sin\sqrt{21}t] $ then the value of k is

If $ f(x)=\sqrt{\frac{1-\cos x}{2}}, 0 < x < 2\pi $ then the value of $ a_0 $ is

Find $ \lim_{z\rightarrow 0} [\frac{1}{1-e^{1/x}} + iy^2] $ if it exists.

Check the continuity of the following function at $ z=0 $:
$ f(z)=\begin{cases} \frac{Re(z^{2})}{|z|^{2}}, & z \ne 0 \\ 0, & z=0 \end{cases} $

Examine whether the Cauchy-Riemann equations are satisfied at the origin for the function given below or not:
$ f(z)=\begin{cases} \frac{x^{3}(1+i)-y^{3}(1-i)}{x^{2}+y^{2}}, & z \ne 0 \\ 0, & z=0 \end{cases} $
Also check, whether $ f(z) $ is analytic at $ z=0 $ or not.

If $ f(\xi)=\int_{C}\frac{3z^{2}+7z+1}{z-\xi}dz, $ where C is an ellipse $ 9x^{2}+16y^{2}=144 $. Find the values of $ f(3) $, $ f'(1-i) $ and $ f''(1-i) $.

Evaluate the integral $ \int_{0}^{\infty}\frac{\sin ax}{x(x^{2}+b^{2})}dx $, $ a>0, b>0 $ by the use of Cauchy residue theorem.

Find the Laurent's expansion for $ f(z)=\frac{7z-2}{z^{3}-z^{2}-2z} $ in the region $ 1 <|z+1|<3 $.

Find the complete solution of $ \frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=x+e^{x}\cos x $.

Find the solution of $ x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+y=\log x \cdot \sin(\log x) $.

Find the solution of $ \frac{d^{2}y}{dx^{2}}-6\frac{dy}{dx}+9y=\frac{e^{3x}}{x^{2}} $ by the method of variation of parameters.

Test the convergence of the series $ \Sigma(1+\frac{1}{n})^{-n^2} $.

Test the convergence of the series $ \frac{a}{b}+\frac{a+x}{b+x}+\frac{a+2x}{b+2x}+\frac{a+3x}{b+3x}+...... $

Find Laplace transform of $ t \cos at $.

Evaluate the integral $ \int_{0}^{\infty}te^{-3t}\sin t dt $, by the use of Laplace transform.

Find the inverse Laplace transform of $ \tan^{-1}(\frac{2}{s^{2}}) $.

Find the solution of the following initial value problem by the use of Laplace transform:
$ \frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}-y=0, $ if $ y(0)=0 $ and $ (\frac{dy}{dt})_{t=0}=1. $

Find the Fourier series expansion of the following function:
$ f(x)=\begin{cases} 0, & -\pi \le x \le 0 \\ \sin x, & 0 \le x < \pi \end{cases} $
Hence deduce that $ \frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{5.7}-\frac{1}{7.9}+......=\frac{\pi-2}{4} $.

Find the Fourier expansion of $ x\sin x $ as Fourier cosine series in the interval $ [0,\pi] $.