Question

3. For each of the following, construct a function $f$, analytic in the plane except for isolated singularities, that satisfies the given conditions. (a) $f$ has a zero of order 2 at $z = i$ and a pole of order 5 at $z = 2 - 3i$. (b) $f$ has a simple zero at $z = 0$ and an essential singularity at $z = 1$. (c) $f$ has a removable singularity at $z = 0$, a pole of order 6 at $z = 1$, and an essential singularity at $z = i$. (d) $f$ has a pole of order 2 at $z = 1 + i$ and essential singularities at $z = 0$ and $z = 1$.

          3. For each of the following, construct a function $f$, analytic in the plane except for
isolated singularities, that satisfies the given conditions.
(a) $f$ has a zero of order 2 at $z = i$ and a pole of order 5 at $z = 2 - 3i$.
(b) $f$ has a simple zero at $z = 0$ and an essential singularity at $z = 1$.
(c) $f$ has a removable singularity at $z = 0$, a pole of order 6 at $z = 1$, and an
essential singularity at $z = i$.
(d) $f$ has a pole of order 2 at $z = 1 + i$ and essential singularities at $z = 0$ and
$z = 1$.

3. For each of the following, construct a function f, analytic in the plane except for
isolated singularities, that satisfies the given conditions.
(a) f has a zero of order 2 at z = i and a pole of order 5 at z = 2 - 3i.
(b) f has a simple zero at z = 0 and an essential singularity at z = 1.
(c) f has a removable singularity at z = 0, a pole of order 6 at z = 1, and an
essential singularity at z = i.
(d) f has a pole of order 2 at z = 1 + i and essential singularities at z = 0 and
z = 1.

Added by Javier C.

Question

Please give Ace some feedback