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A371707 Constant r > 0 satisfying: Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.

%I A371707 #18 Apr 10 2024 12:24:49
%S A371707 2,7,3,4,3,9,0,0,1,9,0,8,5,6,8,3,8,5,5,3,8,7,9,1,7,5,8,0,0,4,6,9,8,1,
%T A371707 5,0,2,4,0,1,7,4,5,5,6,0,1,9,5,3,7,4,0,3,7,9,5,7,8,7,7,4,6,4,5,0,9,3,
%U A371707 5,0,8,6,8,8,7,8,4,2,8,6,6,5,9,7,5,4,3,3,8,7,4,2,2,9,6,2,1,9,5,2
%N A371707 Constant r > 0 satisfying: Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.
%C A371707 Related identity: Sum_{n>=0} (x^n + y)^n/n! = Sum_{n>=0} exp(y*x^n)*x^(n^2)/n!. Here, x = r and y = 2*Pi*i.
%C A371707 What are the roots of Norm( Sum_{n>=0} (x^n + 2*Pi*i)^n/n! ) = 1? The real roots include x = 0 and x = r (this constant).
%H A371707 Paul D. Hanna, <a href="/A371707/b371707.txt">Table of n, a(n) for n = 0..1000</a>
%F A371707 Constant r and related values C and S satisfy the following formulas.
%F A371707 (1) Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.
%F A371707 (2) C = Sum_{n>=0} cos(2*Pi*r^n) * r^(n^2) / n!.
%F A371707 (3) S = Sum_{n>=0} sin(2*Pi*r^n) * r^(n^2) / n!.
%e A371707 The initial 500 digits of this constant r are
%e A371707 r = 0.27343900190856838553879175800469815024017455601953\
%e A371707 74037957877464509350868878428665975433874229621952\
%e A371707 21271807208862504474781327669150216691806622917186\
%e A371707 30052292342530146845288659570856888661537928135397\
%e A371707 91914154858221560663972999347727219299210079054658\
%e A371707 20785838554943078876634169703813817526574697076018\
%e A371707 43103025671330263969269247113168608393647224573552\
%e A371707 82695245129846145197371729802801821910764770241403\
%e A371707 85315562772171090016733480930506290614196661276630\
%e A371707 35680469795753191100711562687066719873558759501438...
%e A371707 Given Sum_{n>=0} (r^n + 2*Pi*i)^n / n! = C + i*S
%e A371707 then C = Sum_{n>=0} cos(2*Pi*r^n) * r^(n^2) / n!, where
%e A371707 C = 0.96236940120128609855708390989630224707797733780139\
%e A371707 33346689286186097367092064030604732865267035268595\
%e A371707 44279783779811281344593178122348416729686502694192\
%e A371707 27215955652725928674242226419071059523037649451781\
%e A371707 36060669147586159699815697962817267659814744582224\
%e A371707 93126268783872251860132042094952557434607056861286\
%e A371707 20902477149931860926346847824008347947488598827305\
%e A371707 47837372109484356517193566333052743194953698066525\
%e A371707 72228584587713864226102674129509160583381421007047\
%e A371707 75118828482389128699072732009353421657729660481717...
%e A371707 and S = Sum_{n>=0} sin(2*Pi*r^n) * r^(n^2) / n!, where
%e A371707 S = 0.27174461472396842102515050866607715426951746748919\
%e A371707 04159412993022348271493896385066506863889535797824\
%e A371707 35085649751784233166430963459007191963331589808443\
%e A371707 52259856849111637575812332490848107413710402589323\
%e A371707 75221334357855133874979455560441735994213395179878\
%e A371707 38917993730963815574520261440791182088848636006332\
%e A371707 68221934823032560291871222621378256174374612116671\
%e A371707 09358271083370500808439006024716239994653435216572\
%e A371707 21204963868973568338610259219318795040671357965613\
%e A371707 68248089245008828798740589773672045329008665505374...
%e A371707 such that C^2 + S^2 = 1.
%Y A371707 Cf. A326600.
%K A371707 nonn,cons
%O A371707 0,1
%A A371707 _Paul D. Hanna_, Apr 09 2024