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Complex Variable
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Q61

Given f(z) = g(z) + h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?

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Q62

Open question

All the values of the multi-valued complex function 1ⁱ, where

i = - 1,

are

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Q63

Open question

If f(x + iy) = x³ - 3xy² + i

ϕ

(x, y) where

i = - 1

and f(x + iy) is an analytic function then

ϕ

(x, y) is

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Q64

Open question

Potential function

ϕ

is given as

ϕ

= x² - y². What will be the stream function

ψ

with the condition

ψ

= 0 at x = y = 0?

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Q65

Open question

Consider likely applicability of Cauchy's Integral Theorem to evaluate the following integral counter clockwise around the unit circle c.

I = c \oint sec z dz,

z being a complex variable. The value of

I

will be

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Q66

Open question

For a complex number z = 1 - 4i with

i = - 1,

the value of

\frac{z + 3}{z - 1}

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Q67

Open question

The value of the contour integral

∣ z - i ∣ = 2 \oint \frac{1}{z ^{2} + 4} dz

in positive sense is

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Q68

Open question

The value of

\oint Γ \frac{3 z - 5}{( z - 1 ) ( z - 2 )} dz

along a closed path

Γ

is is equal to (4πi), where z = x + iy and

i = - 1 .

The correct path

Γ

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Q69

Open question

The contour integral

C \oint e^{\frac{1}{z}} dz

with C as the counter-clockwise unit circle in the z-plane is equal to

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Q70

Open question

A complex function f(z) = u(x, y) + iv(x, y) and its complex conjugate, f'(z) = u(x, y) - iv(x, y) are both analytic in the entire complex plane, where z = x + iy and

i = - 1 .

The function f is then given by

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Q71

Open question

The complex number

(\frac{2 + i}{3 - i})^{2}

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Q72

Open question

The residues of a complex function

X (z) = \frac{1 - 2 z}{z ( z - 1 ) ( z - 2 )}

at its poles are

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Q73

Open question

Which one of the following functions is analytic in the region |z| ≤ 1?

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Q74

Open question

The value of the integral

\oint \frac{2 z + 5}{( z - \frac{1}{2} ) ( z ^{2} - 4 z + 5 )} dz

over the contour |z| = 1, taken in the anti-clockwise direction, would be

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Q75

Open question

An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y), where

i = - 1 .

If u(x, y) = x² - y², then expression for v(x, y) in terms of x, y and a general constant c would be

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Q76

Open question

The residue of the function

f (z) = \frac{1}{( z + 2 ) ^{2} ( z - 2 ) ^{2}}

at z = 2 is

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Q77

Open question

For a complex number z,

z \to i lim \frac{z ^{2} + 1}{z ^{3} + 2 z - i ( z ^{2} + 2 )}

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Q78

Open question

The residue of

f (z) = \frac{z ^{3}}{( z - 1 ) ^{4} ( z - 2 ) ( z - 3 )}

at z = 3 is

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Q79

Open question

Given

f (z) = \frac{1}{z + 1} - \frac{2}{z + 3} .

If C is a counter clockwise path in the z-plane such that |z + 1| = 1, the value of

\frac{1}{2 π j} \oint_{C} f (z) dz

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Q80

Open question

If a complex number ω satisfied the equation ω³ = 1 then value of

1 + ω + \frac{1}{ω}

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Complex Variablepractice.

Complex Variable
practice.