Complex Variable Β· Engineering Mathematics Β· GATE ME
Marks 1
Let f(z) be an analytic function, where z = x + iy . If the real part of f(z) is cosh x cos y , and the imaginary part of f(z) is zero for y = 0 , then f(z) is
The value of k that makes the complex-valued function
π(π§) = πβππ₯ (cos 2π¦ β π sin 2π¦)
analytic, where π§ = π₯ + ππ¦, is _________.
(Answer in integer)
$$f\left( z \right) = u\left( {x,y} \right) + i\,v\left( {x,y} \right).\,$$ If $$u(x,y)=2xy,$$
then $$v(x,y)$$ must be
$$f\left( z \right) = u\left( {x,y} \right) + i\,\,v\,\,\left( {x,y} \right)$$ where $$i = \sqrt { - 1} .$$
If $$u=xy$$ then the expression for $$v$$ should be
Marks 2
If $C$ is the unit circle in the complex plane with its center at the origin, then the value of $n$ in the equation given below is _______ (rounded off to 1 decimal place).
$$ \oint_c \frac{z^3}{\left(z^2+4\right)\left(z^2-4\right)} d z=2 \pi i n $$
The value of the integral
$\rm \oint \left( \frac{6z}{2z^4 - 3z^3 + 7 z^2 - 3z + 5} \right) dz$
evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole z = i, where π is the imaginary unit, is
where $${\rm I} = \sqrt { - 1} ,$$ then
evaluated using contour integration and the residue theorem is
Where $$u(x,y)=2kxy$$ and $$v(x,y)$$ $$ = {x^2} - {y^2}.$$
The value of $$k,$$ for which the function is analytic, is __________.
$$f\left( z \right) = u\left( {x + y} \right) + iv\left( {x,y} \right),$$ where $$i = \sqrt { - 1} .$$ If $$u(x, y)=$$ $${x^3} - {y^2}$$
then expression for $$v(x,y)$$ in terms of $$x,y$$ and a general constant $$c$$ would be