Locate and Classify Singularities

In summary, the function f(z) has singularities at z = i and -i. The singularity at 0 is essential because f(z) is defined in the whole complex plane except for z = 0.
  • #1
jjangub
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0

Homework Statement


Locate and classify the singularities of the following functions
a) f(z) = 1 / (z^3*(z^2+1))
b) f(z) = (1 - e^z)/z
c) f(z) = 1 / (1-e^z(^2))
d) f(z) = z / (e^(1/z))

Homework Equations


The Attempt at a Solution


I am not sure what I need to do when it asks me to locate and classify the singularities. I tried this way.
a) f(z) has singularities when z = i or -i and singularities are fifth order poles.
(z^3*(z^2+1) = z^5 + z^3
b) f(z) has essential singularity at 0 because f(z) is defined in the whole complex plane except for z = 0.
c) f(z) has essential singularity at 0 because f(z) is defined in the whole complex plane except for z = 0.
d) since we know e^(1/z) is essential singularity at 0, f(z) = z / (e^(1/z)) is same.

Did I do right?
Please tell me if I did something wrong.
Thank you.
 
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  • #2
Need to write those in quotient form. It's easier to talk about the singularities that way.

[tex](a)\quad \frac{1}{z^3(z^2+1)}[/tex]

Ok, that one has a third-order pole at zero and simple poles at [itex]\pm i[/itex] right?

[tex](b)\quad \frac{1-e^z}{z}[/tex]

That has a limit as [itex]z\to 0[/itex] so has a removable singularity at zero.

[tex](c)\quad \frac{1}{1-e^{z^2}}[/tex]

That one is tricky because [itex]e^u[/itex] is entire so reaches every value, except maybe one, infinitely often so can you solve multiwise:

[tex]e^{z^2}=1[/tex]

I mean:

[tex]z^2=\log(1)[/tex]

[tex]z=\sqrt{\ln|1|+i(0+2n\pi)}[/tex]

[tex]z=\sqrt{2n\pi i},\quad n=0,\pm 1,\pm2,\cdots[/tex]

So (c) has a pole at those values of z. Keep in mind the square root is double-valued also, so the poles are along the two rays with arguments [itex]\pi/4[/itex] and [itex]-3\pi /4[/itex]. What are the orders of those poles? Can you figure out what the order of the one at zero is if I write it as:

[tex]-\frac{1}{\sum_{n=1}^{\infty} \frac{z^{2n}}{n!}}[/tex]

And (d) is the canonical form of an essential singularity ([itex]e^{1/z}[/itex]). You can write the taylor series for [itex]e^{w}[/itex] and then substitute [itex]w=1/z, z\ne 0[/tex] and obtain a Laurent series with a non-terminating singular part (the part with powers of z in the denominator). Also, [itex]e^u[/itex] is entire and non-polynomial so then has an essential singularity at infinity (Picard) which means [itex]e^{1/z}[/tex] has one at zero.
 
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  • #3
I understand most of them excpet c).
Could you explain again about c)?
Thank you.
 
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Related to Locate and Classify Singularities

1. What is the purpose of locating and classifying singularities?

Locating and classifying singularities is important in understanding the behavior and properties of a system. Singularities, also known as critical points, are points where a function or system experiences a discontinuity or becomes undefined. By identifying and classifying these points, scientists can better understand the overall behavior and stability of the system.

2. How do scientists locate singularities?

Singularities can be located by finding the points where the derivative of a function is equal to zero. This can be done analytically by solving for the roots of the derivative equation, or numerically using mathematical software or algorithms.

3. What are the different types of singularities?

There are three main types of singularities: removable, essential, and infinite. Removable singularities can be "filled in" or removed and do not affect the overall behavior of the system. Essential singularities are non-removable and have a significant impact on the behavior of the system. Infinite singularities occur when the function or system becomes undefined at a certain point.

4. How are singularities classified?

Singularities are classified based on the behavior of the system near the singularity. Removable singularities are classified as either a pole or a removable discontinuity. Essential singularities can be classified as a branch point, an essential singularity, or a non-isolated singularity. Infinite singularities are classified as either a vertical asymptote or a horizontal asymptote.

5. What are the practical applications of locating and classifying singularities?

Locating and classifying singularities has many practical applications in various fields such as physics, engineering, and economics. In physics, understanding the singularities of a system can help predict the behavior of particles or waves. In engineering, identifying and classifying singularities can aid in designing stable systems. In economics, studying singularities can help predict market trends and fluctuations.

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