Expand Expression for Small Values Of …

  • Thread starter Saladsamurai
  • Start date
  • Tags
    Expression
In summary: But I am not sure when I should use the small z/b and small a^2/(bz) assumptions.In summary, the problem statement asks you to expand the log term for small z/b and small a^2/(bz), but does not specify when these assumptions should be used.
  • #1
Saladsamurai
3,020
7
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:

[tex]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}[/tex]

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?
 
Last edited:
Physics news on Phys.org
  • #2
Saladsamurai said:
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.
Saladsamurai said:
I am not so sure how to interpret this? The expression that I am supposed to expand is:

[tex]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}[/tex]

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?
 
  • #3
The first thing I would do is divide every term by the reciprocal, [itex]bz/a^2[/itex], the separate the (b/z) terms- that will, of course have (b/z) to negative powers.
 
  • #4
Mark44 said:
I'm going to interpret your ambiguous expression a2/bz to mean a2/(bz).

Yes, this is correct. (Edited.)

HallsofIvy said:
The first thing I would do is divide every term by the reciprocal, [itex]bz/a^2[/itex], 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:

Screenshot2011-04-20at15428PM.png


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.
 

1. What does "Expand Expression for Small Values Of" mean?

"Expand Expression for Small Values Of" refers to the process of using mathematical techniques to simplify and rewrite a mathematical expression for values that are close to zero. This allows for easier analysis and approximation of the original expression.

2. Why is it important to expand expressions for small values?

Expanding expressions for small values allows for a better understanding and approximation of the original expression, especially in cases where the original expression may be difficult to evaluate or manipulate. It also allows for a better understanding of the behavior of the expression as the values approach zero.

3. What are some common techniques used to expand expressions for small values?

Some common techniques include using Taylor series, binomial expansion, and Maclaurin series. These techniques involve expressing the original expression as a series of terms that become simpler as the values approach zero.

4. Can expanding expressions for small values be applied to any type of mathematical expression?

Yes, expanding expressions for small values can be applied to a wide range of mathematical expressions, including polynomials, trigonometric functions, and exponential functions. However, the specific technique used may vary depending on the type of expression.

5. How does expanding expressions for small values relate to real-world applications?

Expanding expressions for small values has many practical applications, such as in physics, engineering, and economics. For instance, in physics, it can be used to approximate the behavior of a physical system as a value approaches zero. In engineering, it can be used to analyze the stability of a system. In economics, it can be used to approximate the relationship between variables as the values of the variables approach zero.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
531
  • Calculus and Beyond Homework Help
Replies
2
Views
454
  • Calculus and Beyond Homework Help
Replies
3
Views
474
  • Calculus and Beyond Homework Help
Replies
1
Views
456
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
797
  • Calculus and Beyond Homework Help
Replies
2
Views
836
  • Calculus and Beyond Homework Help
Replies
4
Views
900
  • Calculus and Beyond Homework Help
Replies
2
Views
180
  • Calculus and Beyond Homework Help
Replies
3
Views
924
Back
Top