Expand Expression for Small Values Of …

[Saladsamurai]

I am working through a problem from a Fluid Dynamics course and I have gotten to a point on a problem where it says to "… expand the resulting expression for small values of a²/(bz) and for small z/b … "

I am not so sure how to interpret this? The expression that I am supposed to expand is:

$$F(z) = \frac{m}{2\pi}\ln\left [ \frac{(z+b)(z+a^2/b)}{(z-b)(z-a^2/b)}\right ] - \frac{mi}{2}$$

where i is the imaginary number.

Also: do you think it is supposed to say "expand for small z/b" ? Or should it be for small "b/z" ?

I cannot seem to see where any (z/b)'s would come from?

 

[Mark44]

I am working through a problem from a Fluid Dynamics course and I have gotten to a point on a problem where it says to "… expand the resulting expression for small values of a²/bz and for small z/b … "

I'm going to interpret your ambiguous expression a²/bz to mean a²/(bz).

When they say "small z/b" what that means to me is that z << b. Possibly you've seen that notation before. If not, it means that z is very much smaller than b, which would make z/b a very small number. This means you can replace z + b with b, and z - b with -b.

When they say small values of a²/(bz), I interpret this to mean that a²/b << z. This means you can replace z + a²/b with z, and z - a²/b with z as well.

If I'm on the right track here, the log expression simplifies to ln[bz/(-bz)], or ln(-1) assuming that neither b nor z is zero. Since you're working with complex numbers, ln(-1) is defined, one value of which is i(pi), if I'm remembering my complex analysis correctly.

Hope that helps.

I am not so sure how to interpret this? The expression that I am supposed to expand is:

$$F(z) = \frac{m}{2\pi}\ln\left [ \frac{(z+b)(z+a^2/b)}{(z-b)(z-a^2/b)}\right ] - \frac{mi}{2}$$

where i is the imaginary number.

Also: do you think it is supposed to say "expand for small z/b" ? Or should it be for small "b/z" ?

I cannot seem to see where any (z/b)'s would come from?

 

[HallsofIvy]

The first thing I would do is divide every term by the reciprocal, $bz/a^2$, the separate the (b/z) terms- that will, of course have (b/z) to negative powers.

 

[Saladsamurai]

I'm going to interpret your ambiguous expression a²/bz to mean a²/(bz).

Yes, this is correct. (Edited.)

The first thing I would do is divide every term by the reciprocal, $bz/a^2$, the separate the (b/z) terms- that will, of course have (b/z) to negative powers.

Before I saw this post divided everything by z and ended up with the following:

I am just not sure when I am supposed to use the small z/b and small a^2/(bz) assumptions? I am pretty sure the "expand the resulting expression for small values ..." part of the problem statement means to use a Series expansion of the log term.