What are some examples of functions of x and y or complex functions

In particular what are some functions whose set of zeroes or poles is of measure >0?

Well?

Surely there are such functions?

>>8014693
Ayy what's the value of a contour integral around Warsaw?
It diverges because the Poles are dense

>>8016170
What are examples of functions with dense sets of zeroes or poles?
What about uncountable sets of zeroes or poles?

The sum of all z^n for z >1

>>8014693
>an uncountable set of poles
Trying to create NP-hard problems, OP?

ln(x+y-1)=z

Go back to middle school.

What are some functions that act like the one in the OP image?

>>8016853
Look up implicit equations and take multiplicative inverses of everything you find.

>>8016712
See? It wasn't NP-hard, just plain old implicit equations.

>>8016853
Integral transforms are a good way to spot unbounded behavior of a time-domain system.
To answer the question: the solution you'll get from such ITFs.
Poles are pretty rad in regard of signal or control theory.

Isn't there something really pleasant about girthy poles?

Don't you think it's interesting to have uncountable sets of poles?

Now what are some complex functions that act like this?

>>8020665
Is it difficult to construct complex functions with uncountable zeros/poles?

Surely there are complex functions that diverge at uncountably many points? Possibly so that a closed path of poles confines a region of points of the domain at which the function is undefined?

Shouldn't this be fairly elementary and straightforward for /sci/?

Are there only complex functions with isolated zeros / poles? If so, why?

>>8020665
>>8022179
>>8023423
>>8023513
>>8023840
Do you understand the question you're asking? If so, you can answer it on your own. If you don't, we can begin investigating your motives.

>>8023900
>If so, you can answer it on your own.
How come?

>>8014693
Maybe try log functions that are translated for the singularity to be over the region not defined

Is it possible to construct a complex function that is zero along the perimeter of the unit circle?

Why is it that cluster points and natural boundaries seem to be the closest we can get to such behaviour with complex functions?

cosec(1/x) has an infinite number of poles about the origin, would it be a countably infinite set?

>>8024805
Well it's the closes you can get with analytic functions. You could define a function 1/f(x) where f(x) smoothly reaches zero and remains at zero for some range but it wouldn't be analytic. There's a rule saying that if an analytic function is equal to another analytic function for a continuous range of values then both the functions are equal everywhere and are therefore the same function, a function which is zero for some range would then either be zero everywhere or non-analytic.

>>8016689
>The sum of all z^n for z >1
The resulting surface is thin in comparison to the surrounding space, in contrast to a parabola that eventually takes up half of the surrounding space.
What are some ways of expressing the difference between these two types of surfaces?