5.

Velocity vector [eqn: v] for a fluid flow around a solid sphere of radius [eqn: a] is given by [eqn: v = TRIANGLED PHI]

[eqn: theta]

[eqn: a]

where [eqn: PHI (`r,` theta `)` BETA `] the velocity potential satisfying

[eqn: NABLA ^{2} PHI `=`0]  ( Laplace equation )

Obtain the velocity potential [eqn: PHI `(`r,` theta `)] satisfying the following boundary conditions :

[eqn: {delta PHI } over {delta r} `(`a,` theta `)`=`0]  ( no slip BC )

[eqn: {delta PHI } over {delta r} `(` INF ,` theta `)`=`v _{0} `cos theta] ( [eqn: v _{0}] = constant )

[ Hint ]

① Use both growing and decaying harmonics and apply the BCs

② Legendre polynomials :

[eqn: P _{0} (`cos theta `)`=`1# P _{1} (`cos theta `)`=`cos theta # P _{2} (`cos theta `)`=` {1} over {2} `(`3`cos ^{2} theta `-`1`)]

6.

What kind of poles or essential singularity do the following equation have?

1) [eqn: f(z)`=` {e ^{z} -1} over {z ^{2}}]

2) [eqn: g`(z)`=` {sin`z} over {z ^{2}}]

7.

Find all Taylor and Laurent series of [eqn: f`(z)`=` {1} over {1-z ^{2}}]  with center z=1

8.

Evaluate the following integrals

1) [eqn: int _{0} ^{2 pi } {{d theta } over {1`+` varepsilon `cos theta } ```````(` LEFT | varepsilon RIGHT | `<`1)}]

2) [eqn: int _{- INF } ^{INF } {{cos`( {pi x} over {2} )} over {1`-`x ^{2}} `dx}]