e^(-1/z^2) Essential Singularity

[@Mjafar2000]

What is zero of this function??

[@shubh.ruhela]

Maine pada tha tha ki function exp(-1/z^m) nowhere analytic hota hai. Nowhere analytic hai to no singularities nahi hogi yaha ??? Plz clear my confusion!

[@shubh.ruhela]

@@prachi_ but this opposes the definition of singularity! Csir Feb 2022 QID 527 mein ek aisa question aaya tha.

[@prachi_]

@@shubh.ruhela ok i got your point. According to the definition of singularity, a point is said to be a singularity of a function if its neighbourhood contains at least 1 point at which it is analytic .

[@shubh.ruhela]

@@prachi_ now can we can the given function is nowhere analytic but satisfies the CR equation over C ? Also, what if the function is not given in the price wise form ( i.e. f(z) = 0 at z =0 not given) only f(z) = exp(-1/z²) is given then what can be about its singularities? I found two answers of this question 1) z=0 is an essential singularity (coaching institute assignment sheet) 2) no singularities (in official answer key and in dips academy handwritten notes) Plz clear my doubt! Which one is right ?

[@prachi_]

If the function is not analytic it will not satisfy CR equation

[@prachi_]

And at z=0 the f(z) = e^ (-1/z^2) will give 0 . Whether it is mention or not mention in question

[@Amitkumar-vk3ln]

nice explanation

[@prachi_]

🙂