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Recent questions in Analysis of complex variables

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3
GATE2020: 15
Let $f(z)=\frac{1}{z+a},a>0.$ the value of the integral $\oint f(z)dz$ over a circle $C$ with center $(-a,0)$ and radius $R>0$ evaluated in the anti-clockwise direction is ____________ $0$ $2\pi i$ $-2\pi i$ $4\pi i$
Let $f(z)=\frac{1}{z+a},a>0.$ the value of the integral $\oint f(z)dz$ over a circle $C$ with center $(-a,0)$ and radius $R>0$ evaluated in the anti-clockwise direction i...
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4
GATE2017: 3
Let $z=x+jy$ where $j=\sqrt{-1}$. Then $\overline{\cos z}$ = $\cos z$ $cos\overline{z}$ $\sin z$ $\sin\overline{z}$
Let $z=x+jy$ where $j=\sqrt{-1}$. Then $\overline{\cos z}$ =$\cos z$$cos\overline{z}$$\sin z$$\sin\overline{z}$
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6
GATE2016-5
In the neighborhood of $z=1$, the function $f(z)$ has a power series expansion of the form $f(z)$ = $1$ + $(1-z)$ + $(1-z)^2+ \ldots$ Then $f(z)$ is $\frac{1}{z}$ $\frac{-1}{z-2}$ $\frac{z-1}{z+}$ $\frac{1}{2z-1}$
In the neighborhood of $z=1$, the function $f(z)$ has a power series expansion of the form $f(z)$ = $1$ + $(1-z)$ + $(1-z)^2+ \ldots$Then $f(z)$ is$\frac{1}{z}$$\frac{-1}...
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7
GATE2016-30
The value of the integral $\displaystyle{}\frac{1}{2\pi j}\int_c \frac{Z^2+1}{Z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in the complex plane is $\_\_\_\_\_\_\_\_.$
The value of the integral $\displaystyle{}\frac{1}{2\pi j}\int_c \frac{Z^2+1}{Z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in th...
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8
GATE2015-12
The value of $\oint \frac{1}{Z^2} dZ,$ where the contour is the unit circle traversed clockwise, is $-2\pi i$ $0$ $2\pi i$ $4\pi i$
The value of $\oint \frac{1}{Z^2} dZ,$ where the contour is the unit circle traversed clockwise, is$-2\pi i$$0$$2\pi i$$4\pi i$
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