Help me parse the logic of this statement

  • Thread starter Thread starter Poopsilon
  • Start date Start date
  • Tags Tags
    Logic
AI Thread Summary
The discussion revolves around understanding the statement regarding the residue of an analytic function at a singularity. The key points include the need to determine what can be assumed about the function f and what must be proven, specifically whether the statement is biconditional or conditional. Participants suggest that two main aspects need to be proven: the uniqueness of the residue as a complex number and the existence of a primitive for the modified function in a punctured neighborhood. The original poster expresses frustration in clarifying these logical components and seeks additional perspectives. The conversation highlights the complexity of analyzing analytic functions and their residues in complex analysis.
Poopsilon
Messages
288
Reaction score
1
So I have this statement that I'm supposed to prove and I cannot for the life of me figure out what parts I'm allowed to assume and what part I am expected to prove, here it is:The residue of an analytic function f at a singularity a ∈ ℂ is the uniquely determined complex number c, such that the function
f(z) - \frac{c}{z-a}
admits a primitive in a punctured neighborhood of the point a. (end statement)I know I'm allowed to assume that f is analytic with a singularity at a, but beyond that I just can't tell if it's a biconditional I have to prove, or if it's just a conditional and if so which way. Thanks
 
Mathematics news on Phys.org
I think if means prove two things.

1. The residue is a uniquely determined complex number c.
2. f(x) has the property stated.
 
Thanks for your input Aleph, but it's not the answer I want to hear lol. Can I get a second opinion?
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »
Thread 'Imaginary Pythagorus'

Thread 'Imaginary Pythagorus'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

I Analytic functions of analytic functions
Replies
3
Views
2K
I Contrapositive of theorem and issue with proof
Replies
5
Views
2K
A Question about proof of Great Picard Theorem
Replies
3
Views
2K
I Is the negation of the statement in the book correct too?
Replies
11
Views
1K
I Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
Replies
7
Views
4K
I Can chatgpt accurately calculate expected lengths in Pascal's triangle?
2
Replies
66
Views
6K
Show that the real part of a certain complex function is harmonic
Replies
7
Views
2K
I Expansion of a complex function around branch point
Replies
3
Views
2K
Compute the residue of a function
Replies
2
Views
2K
Need help with finding residue of a simple function
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top