Complex analysis quiz

The function $w\left(z\right)=\frac{1}{\left(z-a\right)}=u\left(x,y\right)+jv\left(x,y\right)$ . Select the most correct statement from the list of options below:

The real functions u(x, y) and v(x, y) have a pole singularity only on the x = Re(z) axis, but the cauchy-riemann equations are satisfied at all other points excluding the singularity.

The functional independence of dw/dz over $\frac{\delta z}{\delta x}$ and/or $\frac{\delta z}{\delta y}$ , where z = x + jy, forms the algebraic basis of deriving the cauchy-riemann equations

The statement is correct, since the uniqueness of the derivative dw/dz is implicitly enforced by requiring its independence from $\frac{\delta z}{\delta\left(x,y\right)}$

The binomial theorem is frequently used to derive the Laurent series for a function like 1/(z-p), that results in two different series forms of the same function. Give this information, select the best answer from the list of choices below.

The two series converges in regions defined by |z| → 0 and |z| →$\infty$ , but, they are discontinuous at z = p

Select the most correct response from the list of statements below

cauchy integral theorem, i.e., $\int f\left(z\right)dz=0$ , stipulates that f(z) must be an analytic function and its component real functions must have continuous partial derivatives on, and within the region formed by the contour C.

Select the most correct statement from the list of choices below

Cauchy integral formula, i.e., $\int\frac{f\left(z\right)}{z-a}dz=2\pi jf\left(a\right)$ , requires the contour C to enclose the point z = a, and f(z) must be analytic function

The function $\frac{1}{\left(z^2+a^2\right)\left(z+3\right)}$ is resolved into partial fractions. Select the most correct statement from the list of options below:

There are three distinct terms in the partial fraction expansion with poles on the real and imaginary axis

The function $w\left(z^{\cdot}\right)=u\left(x,y\right)+jv\left(x,y\right)$ was integrated along a contour in the complex z-plane with no pole singularities enclosed. The result was non-zero. Select the most correct statement from the list of options below.

Perhaps $w\left(z^{\cdot}\right)$ is not analytic anywhere in the entire complex plane that is the underlying reason for the discrepancy

The Laurent series for $\frac{1}{z-1}$ in the region |z| > 1 was obtained. Select the most correct statement from the list of options below.

The Laurent series would possibly have the form $\frac{a-1}{\left(z-1\right)}+a_0+a_1\left(z-1\right)+a_2\left(z-1\right)^2+\cdots$

The analytic function w(z) = u(x, y) + jv(x, y) generates two different gradients U and V. Select the most correct statement from the list of options below.

The gradients obey the relationship u * v = 0