MaplePrimes - comments on Post, MRB Constant K

MaplePrimes - comments on Post, MRB Constant K http://www.mapleprimes.com/posts/87683-MRB-Constant-K en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Sun, 14 Jun 2026 00:50:42 GMT Sun, 14 Jun 2026 00:50:42 GMT The latest comments added to the Post, MRB Constant K http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, MRB Constant K http://www.mapleprimes.com/posts/87683-MRB-Constant-K Therom MRBK 1.0 http://www.mapleprimes.com/posts/87683-MRB-Constant-K?ref=Feed:MaplePrimes:MRB Constant K:Comments#comment88434   If a function is differentiable at some point c of its domain, then it is also continuous at c. However here we extend the notion of differentiability to be valid for individual points on the real number line, specifically positive integers.   DEFINITIONS f(n)=(-1)^n* n^(1/n) g(n)= n^(1/n) k(n)' =d/dn k(n) ∈ means in. n  In the set of integers is n ∈ {1,2,3,...}. Re(f(n)) is the real part of f(n)     THEROM MRBK 1.0                                                                                   .   Re(((-1)^n*n^(1/n))')=(-1)^n*(n^(1/n))' | n ∈ {1,2,3,...}, The real part of the instantaneous change of f relative to n are the same as the instantaneous change of g relative to n. Proof: restart simplify(`assuming`([Re(diff((-1)^n*n^(1/n), n))-(-1)^n*(diff(n^(1/n), n))], [n::posint])) 0  COROLLAY MRBK 1.1  When n is in the set of integers the derivative of f  has no imaginary part. The latest comments added to the Post, MRB Constant K 88434 Mon, 24 May 2010 04:32:41 Z Marvin Ray Burns Marvin Ray Burns The corollary is wrong http://www.mapleprimes.com/posts/87683-MRB-Constant-K?ref=Feed:MaplePrimes:MRB Constant K:Comments#comment88709 The corollary is wrong. Proof. <pre>f:=(-1)^n* n^(1/n); n (1/n) f := (-1) n diff(f,n); n (1/n) n (1/n) / ln(n) 1 \ (-1) Pi n I + (-1) n |- ----- + ----| | 2 2 | \ n n / </pre> Alec The latest comments added to the Post, MRB Constant K 88709 Fri, 28 May 2010 05:07:02 Z Alec Mihailovs Alec Mihailovs COROLLARY MRBK 1.1a http://www.mapleprimes.com/posts/87683-MRB-Constant-K?ref=Feed:MaplePrimes:MRB Constant K:Comments#comment88710 You are correct Alec.  I'm sorry that I let the misreading of my own notation lead me astray.  COROLLAY MRBK 1.1a  When n is in the set of integers the derivative of f has an imaginary part of Pi*f. The latest comments added to the Post, MRB Constant K 88710 Mon, 31 May 2010 02:28:16 Z Marvin Ray Burns Marvin Ray Burns THEOREM MRBK 3.0 http://www.mapleprimes.com/posts/87683-MRB-Constant-K?ref=Feed:MaplePrimes:MRB Constant K:Comments#comment88712     A reminder: If a function is differentiable at some point c of its domain, then it is also continuous at c. However here we extend the notion of differentiability to be valid for individual points on the real number line, specifically positive integers.   f(n)=(-1)^n* n^(1/n) THEOREM MRBK 3.0 When n is in the set of integers the derivative of f has a real part of exactly (1-ln(n))/n^2*f. Proof: > simplify(`assuming`([Re(diff(f(n), n)-(1-ln(n))*f(n)/n^2)], [n::posint])); 0   The latest comments added to the Post, MRB Constant K 88712 Mon, 31 May 2010 05:26:44 Z Marvin Ray Burns Marvin Ray Burns THEOREM MRBK 4.0 http://www.mapleprimes.com/posts/87683-MRB-Constant-K?ref=Feed:MaplePrimes:MRB Constant K:Comments#comment88713   f(n)=(-1)^n* n^(1/n) THEOREM MRBK 4.0 When n is in the set of integers the derivative of f is exactly I*Pi*f+(1-ln(n))*f/n^2 Proof: THEOREM MRBK 2.0 says the Imaginary part is Pi*f. THEOREM MRBK 3.0 says the real part is (1-ln(n))*f/n^2. A complex number is the sum of its imaginary part and its real part.     The latest comments added to the Post, MRB Constant K 88713 Mon, 31 May 2010 05:27:21 Z Marvin Ray Burns Marvin Ray Burns THEOREM MRBK 2.0 http://www.mapleprimes.com/posts/87683-MRB-Constant-K?ref=Feed:MaplePrimes:MRB Constant K:Comments#comment88711 A reminder: If a function is differentiable at some point c of its domain, then it is also continuous at c. However here we extend the notion of differentiability to be valid for individual points on the real number line, specifically positive integers. COROLLARY MRBK 1.1a doesn't seem to follow readily enough so I will make it a theorem.   DEFINITIONS f(n)=(-1)^n* n^(1/n) g(n)= n^(1/n) k(n)' =d/dn k(n) ∈ means in. n  In the set of integers is n ∈ {1,2,3,...}. Im(f(n)) is the imaginary part of f(n) f := proc (n) options operator, arrow ... end proc; is f:=n-> THEOREM MRBK 2.0   When n is in the set of integers the derivative of f has an imaginary part of exactly Pi*f. Proof: > f := proc (n) options operator, arrow; (-1)^n*n^(1/n) end proc; simplify(`assuming`([Im(diff(f(n), n))-Pi*f(n)], [n::posint])); 0       The latest comments added to the Post, MRB Constant K 88711 Mon, 31 May 2010 05:31:12 Z Marvin Ray Burns Marvin Ray Burns