Calculating Limits of Complex Functions

In summary: If you think you have b), can you show us your work?In summary, for a), you need to use the substitution of exp(ia) = cos a - isin a, and for b), you need to use the complex conjugate of z.
  • #1
Mattofix
138
0

Homework Statement



Compute limit as n-> infinity of

a)zn = exp(in^2)/(1+in^2)

b)zn = 1/(n + i)

Homework Equations



?

The Attempt at a Solution



These are 2 examples of a series of questions i have to complete. I can see that i need to calculate the limit for a complex function but i have not come across this in lectures yet. Could you please point me in the right direction, maybe i need to use a certain rule/law/method?
 
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  • #2
Hint: look at the moduli first, that is, consider
[tex]\lim_{n \to \infty} |z_n|[/tex]
 
  • #3
Finding the limit in the complex numbers is pretty much the same as real numbers. As compuchip said, you need to look at [itex]|z_n|= \sqrt{(x_n)^2+ (y_n)^2}[/itex]. Although it won't give you the limit exactly, you should try calculating a few terms of the sequence to get an idea of what's happening. Have you done that?
 
  • #4
should i be using the subtitution of exp(ia) = cos a - isin a ? or is there a better method?
 
  • #5
yeah - i know that it tends to 0 - but I am not too sure about what i use for the xn and yn parts and what that eventually gives me - if i can use exp(ia) = cos a - isin a i think i have crcked it?
 
  • #6
You can always write it out in real and imaginary parts and notice that
[tex]\lim_{z \to 0} f(z) = \left( \lim_{z \to 0} \operatorname{Re}(f(z)) \right) + \mathrm{i} \left( \lim_{z \to 0} \operatorname{Im}(f(z)) \right)[/tex].
Then you can indeed use exp(i a) = cos(a) + i sin(a) for the first one, and use that the real and imaginary parts of a fraction can be determined by writing
[tex]\frac{\alpha}{z} = \frac{\alpha}{z} \frac{\bar z}{\bar z}[/tex]
with [itex]\bar z[/itex] the complex conjugate of z.

So if you think you have a), can you show us your work?
 
Last edited:

1. What is the definition of a limit for a complex function?

The limit of a complex function is the value that the function approaches as the input values get closer and closer to a particular point.

2. How do you calculate the limit of a complex function algebraically?

To calculate the limit of a complex function algebraically, you can use techniques such as factoring, rationalizing, or expanding to simplify the function and then plug in the desired value for the input variable.

3. What is the difference between a one-sided limit and a two-sided limit for a complex function?

A one-sided limit only considers the behavior of the function as the input values approach the desired point from one direction (either from the left or the right), while a two-sided limit takes into account the behavior from both directions.

4. Can the limit of a complex function exist at a point where the function is not defined?

No, the limit of a complex function can only exist at a point where the function is defined. If the function is not defined at a particular point, the limit at that point does not exist.

5. How can graphical representations help in understanding limits of complex functions?

Graphs can help visualize the behavior of a complex function and its limit at a particular point. They can also aid in identifying any discontinuities or asymptotes that may affect the existence of the limit.

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