Poles or Removable Singularities

In summary, the conversation discusses finding the location and nature of singularities in a given function and applying the Cauchy Integral Formula. The function in question has a pole of order 14 at z=0 and when applying the Cauchy Integral Formula, the derivative of the function is taken and multiplied by 2 pi j and 1/14!. This results in a value of 0.
  • #1
Chris0724
2
0

Homework Statement



Determine the location and nature of singularities in the finite z plane of the follow function and apply Cauchy Integral Formula


Homework Equations



g(z) =

sin 2z
-------
z^15


The Attempt at a Solution



I know there is a pole of order 14 at z = o

but I'm a bit confuse when i apply the Cauchy Integral Formula

{sin 2z } / z
------------
```z^15

= 2 pi j { d ^13 f(z) / dz}
```````-----------------
`````````````13!

= 2 pi j / 13! <-- correct ?

many thanks! :)
 
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  • #2
You'd better take another careful look at the Cauchy integral theorem. I don't know where you got the '13' and what is that f(z) that you are taking the 13th derivative of? Why did it just disappear?
 
  • #3
Dick said:
You'd better take another careful look at the Cauchy integral theorem. I don't know where you got the '13' and what is that f(z) that you are taking the 13th derivative of? Why did it just disappear?

hi,

i think i make a mistake...

sin 2z
-------
z^15

since sin 2z is analytic, f(z) = sin 2 z

2 pi j { d^14 f(z) / dz }

= 2 pi j { 0 }

= 0 <-- correct ?
 
  • #4
Chris0724 said:
hi,

i think i make a mistake...

sin 2z
-------
z^15

since sin 2z is analytic, f(z) = sin 2 z

2 pi j { d^14 f(z) / dz }

= 2 pi j { 0 }

= 0 <-- correct ?

That's better. You are missing a 1/14! But it doesn't matter because the derivative is zero anyway.
 

1. What are poles or removable singularities?

Poles and removable singularities are types of discontinuities in a function. They occur when a function approaches a particular point but does not exist at that point.

2. How do poles or removable singularities differ from each other?

The main difference between poles and removable singularities is that poles are essential singularities, meaning the function approaches infinity at that point, while removable singularities can be "fixed" by redefining the function at that point.

3. What causes poles or removable singularities?

Poles are caused by division by zero, while removable singularities can be caused by a variety of factors such as the cancellation of common factors in a rational function or the presence of a removable discontinuity in a function.

4. How are poles or removable singularities useful in mathematics?

Poles and removable singularities are useful in understanding the behavior of functions and their limits. They also have applications in complex analysis, where they are used to study the behavior of complex functions.

5. Can poles or removable singularities be avoided when graphing a function?

In some cases, poles and removable singularities can be avoided by redefining the function or using a different representation. However, in other cases, they are inherent to the function and cannot be avoided.

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