Complex Variables and Partial Differential Equations (3130005) MCQ

MCQs of Laurent's Series & Residues

MCQ No - 1

The Laurent series of function $$\mathrm f\left(\mathrm z\right)=\frac1{\mathrm z\left(\mathrm z-1\right)}$$ for the region $$\left|\mathrm z\right|<1$$ is _____ .

(A)

$$1-\mathrm z-\mathrm z^2-......$$

(B)

$$\frac1{\mathrm z}+1+\mathrm z+\mathrm z^2-......$$

(C)

$$-\frac1{\mathrm z}-1-\mathrm z-\mathrm z^2-......$$

(D)

none of these


C
MCQ No - 2

The Laurent series of function $$\mathrm f\left(\mathrm z\right)=\frac1{\mathrm z\left(\mathrm z-1\right)}$$ for the region $$\left|\mathrm z-1\right|<1$$ is _____ .

(A)

$$\frac1{\mathrm z-1}-1+(\mathrm z-1)-{(\mathrm z-1)}^2+......$$

(B)

$$\frac1{\mathrm z-1}+1+(\mathrm z-1)+{(\mathrm z-1)}^2+......$$

(C)

$$\frac1{\mathrm z-1}-1+z-\mathrm z^2+......$$

(D)

none of these


A
MCQ No - 3

The principal part of Laurent series of $$\mathrm f\left(\mathrm z\right)=\frac{\mathrm e^{\mathrm z}-1}{\mathrm z^2}$$ is _____ .

(A)

$$z$$

(B)

$$\frac1{\mathrm z}$$

(C)

$$\frac1{\mathrm z^2}$$

(D)

none of these


B
MCQ No - 4

The laurent series of function $$\mathrm{zcos}\left(\frac1{\mathrm z}\right)$$ about $$z=0$$ is _____ .

(A)

$$\mathrm z+\frac1{2!}\frac1{\mathrm z}+\frac1{4!}\frac1{\mathrm z^3}+\frac1{6!}\frac1{\mathrm z^5}+......$$

(B)

$$\mathrm z-\frac1{2!}\frac1{\mathrm z}+\frac1{4!}\frac1{\mathrm z^3}-\frac1{6!}\frac1{\mathrm z^5}+......$$

(C)

$$\mathrm z-\frac12\frac1{\mathrm z}+\frac14\frac1{\mathrm z^3}-\frac16\frac1{\mathrm z^5}+......$$

(D)

none of these


B
MCQ No - 5

The residue of $$\mathrm f\left(\mathrm z\right)=\frac{\mathrm e^{2\mathrm z}}{\mathrm z-2}$$ is _____ .

(A)

$$\mathrm e$$

(B)

$$\mathrm e^2$$

(C)

$$\mathrm e^3$$

(D)

$$\mathrm e^4$$


D