Blissard's trigonometric series with closed-form sums
aa r X i v : . [ m a t h . HO ] F e b BLISSARD’S TRIGONOMETRIC SERIES WITH CLOSED-FORM SUMS
JACQUES G´ELINAS
Abstract.
This is a summary and verification of an elementary note written by John Blissardin 1862 for the Messenger of Mathematics. A general method of discovering trigonometric serieshaving a closed-form sum is explained and illustrated with examples. We complete some state-ments and add references, using the summation symbol and Blissard’s own (umbral) representativenotation for a more concise presentation than the original. More examples are also provided. Historical examples
Blissard’s well structured note [3] starts by recalling four trigonometric series which “mathemati-cal writers have exhibited as results of the differential and integral calculus” (A, B, C, G in the tablebelow). Many such formulas had indeed been worked out by Daniel Bernoulli, Euler and Fourier.More examples can be found in 19th century textbooks and articles on calculus, in 20th centurytreatises on infinite series [21, 10, 19, 17] or on Fourier series [12, 23], and in mathematical tables[18, 22, 16, 11].G.H. Hardy motivated the derivation of some simple formulas as follows [17, p. 2]. We can firstagree that the sum of a geometric series 1 + x + x + . . . with ratio x is s = 1 / (1 − x ) because this istrue when the series converges for | x | <
1, and “it would be very inconvenient if the formula variedin different cases”; moreover, “we should expect the sum s to satisfy the equation s = 1 + sx ”.With x = e iθ , we obtain immediately a number of trigonometric series by separating the real andimaginary parts, by setting θ = 0, by differentiating, or by integrating [14, §
13 ]. The classicaltheory of Fourier series [12] can next give a rigorous proof of our conjectured formulas with theirdomain of convergence, and can also give more sums.Abel’s limit theorem [19, p. 154, 405] provides a very simple proof in some cases. Indeed, Abel’ssummation by parts shows that if a real power series f ( r ) = P a k r k with radius of convergenceequal to 1 converges for r = 1, then f ( r ) is continuous from the left at r = 1. It is thus sufficient tofind a closed-form expression for f ( r ) and take its limit as r → − in order to justify (or discover)a closed-form formula for f (1). For example, if 0 < r < < θ < π , then [21, p. 211] ∞ X n =1 (cid:0) re iθ (cid:1) n n = Z re iθ dt − t = − log (cid:0) − re iθ (cid:1) = − log (1 − r cos θ − ir sin θ )= −
12 log (cid:18) (1 − r ) + 4 r sin θ (cid:19) + i arctan (cid:18) r sin θ − r cos θ (cid:19) . Now Abel’s test of convergence for power series with positive coefficients decreasing to 0, proven by(1 − z ) P nk =0 a k z k = a + a n z n +1 + P nk =1 ( a k − a k +1 ) z k , shows that P e inθ /n converges if e iθ = 1[10, p. 244]. Abel’s limit theorem can thus be applied here, and by taking real and imaginary partsin Euler’s formula for e inθ we fully justifiy the first two equations (A, B) in the following table, andthe next six formulas (C,D,E,F,G,H) follow immediately. Recent derivations of (B) can be found in[20] (Fourier series) and in [13] (Laplace transforms). Mathematics Subject Classification.
Primary 42A32, Secondary 05A40.
Key words and phrases.
Trigonometric series, Blissard symbolic notation, umbral calculus.This work was done in 2019 while the author was a retired mathematician.
Equation Domain MethodA ∞ X n =1 cos nθn = − log (cid:18) θ (cid:19) < θ < π ∞ X n =1 (cid:0) re iθ (cid:1) n n , | r | ≤ ∞ X n =1 sin nθn = π − θ < θ < π [21, p. 211]C ∞ X n =1 ( − n − cos nθn = log (cid:18) θ (cid:19) − π < θ < π θ → π − θ in AD ∞ X n =1 ( − n − sin nθn = θ − π < θ < π θ → π − θ in BE ∞ X n =0 cos(2 n + 1) θ n + 1 = −
12 log (cid:18) tan θ (cid:19) < θ < π A + CF ∞ X n =0 sin(2 n + 1) θ n + 1 = π < θ < π B + DG ∞ X n =0 ( − n cos(2 n + 1) θ n + 1 = π − π < θ < π θ → π − θ in FH ∞ X n =0 ( − n sin(2 n + 1) θ n + 1 = 14 log 1 + sin θ − sin θ − π < θ < π θ → π − θ in EI ∞ X n =1 cos 2 nπθ (2 nπ ) k = ( − k − B k ( θ )(2 k )! 0 ≤ θ ≤ k = 1 , , . . . [9, p. 256] , [10, p. 370]J ∞ X n =1 sin 2 nπθ (2 nπ ) k +1 = ( − k − B k +1 ( θ )(2 k + 1)! 0 < θ < k = 00 ≤ θ ≤ k = 1 , , . . . [9, p. 256] , [10, p. 370]K ∞ X n = −∞ cos( n − a ) θn − a = − π tan( aπ ) 0 < θ < π ; a ∈ R [9, p. 230] , [10, p. 371]L ∞ X n = −∞ sin( n − a ) θn − a = π < θ < π ; a ∈ R [9, p. 230] , [10, p. 371]M ∞ X n =1 ( − n +1 cos nθn = 7 π − π θ
24 + θ − π ≤ θ ≤ π k = 2 , θ → θ π + 12 in IN ∞ X n =1 ( − n +1 sin nθn = π θ − θ − π ≤ θ ≤ π nθ = 1 − cos 2 nθ LISSARD’S TRIGONOMETRIC SERIES 3 Binomial summation formula
Blissard first proves a summation formula involving a positive integer n and a parameter m ,which he also used later to derive a formula involving Bernoulli numbers [4, p. 56]. Lemma 2.1. n X k =0 ( − k (cid:18) nk (cid:19) m + k = n ! Q nk =0 ( m + k ) , ( m > , n = 0 , , , . . . ) . Proof.
With Blissard’s representative notation, let R k = 1 /k , and downgrade exponents into indicesafter expansion . It is thus required to prove R m (1 − R ) n = n ! Q nk =0 R m + k , which is an identity if n = 0. If we assume that this has been proven for a certain integer n ≥ m >
0, then R m (1 − R ) n +1 = R m (1 − R ) (1 − R ) n = R m (1 − R ) n − R m +1 (1 − R ) n = n ! n Y k =0 R m + k − n ! n Y k =0 R m +1+ k = (cid:0) R m − R m + n +1 (cid:1) n ! n Y k =1 R m + k = ( n + 1) R m R m + n +1 n ! n Y k =1 R m + k (cid:18) m − m + n + 1 = n + 1 m ( m + n + 1) (cid:19) = ( n + 1)! n +1 Y k =0 R m + k . Thus the identity in question holds for the next integer n + 1 and all m > (cid:3) Since the same term 1 /m appears on both sides, the identity can be said to hold also as m → n X k =0 ( − k (cid:18) nk (cid:19) m + k = Z x m − (1 − x ) n dx = x m m (1 − x ) n (cid:12)(cid:12)(cid:12)(cid:12) + nm Z x m (1 − x ) n − dx = n ( n − m ( m + 1) Z x m +1 (1 − x ) n − dx = . . . = Γ( n + 1)Γ( m )Γ( m + n + 1) . Boole [8, p. 26] obtains this from the n -th difference of 1 /m = R ∞ e − mx dx , a method of Abel, andone can also use differences of falling factorial powers [15, p. 188] or partial fraction decomposition.3. Maclaurin logarithm expansion
A second lemma uses the first one to obtain the expansion of a binomial polynomial times alogarithm via the Cauchy product of absolutely convergent power series (we omit the proof). The A n,k will appear in the examples presented below. Lemma 3.1. If n is a positive integer and | x | < , then (1 + x ) n log(1 + x ) = n X k =1 A n,k x k + n ! ∞ X k =1 ( − k − ( k − n + k )! x n + k , where A n,k := k X j =1 ( − k − j (cid:18) nj − (cid:19) k − j + 1 . R k := R k for k >
0. We use R m R n = R m + n and R m C − R n C = ( R m − R n ) C for m > , n > JACQUES G´ELINAS The general recipe
A simple method for discovering trigonometric series with a closed-form sum is to start from theMaclaurin expansion of a known function of x and set x = e iθ , as was done above for the geometricseries. Blissard states that this works, “whatever form f ( x ) may assume”, for the expressions f ( x ) ± f ( x − ) which “can be evaluated in terms of trigonometrical functions of θ ” with the help ofthe lemma below. Of course, these combinations give twice the real and imaginary parts of f ( x ) if f is real. We again omit the proof, done by a simple verification of each case. Lemma 4.1. If x = e iθ and cis θ := cos θ + i sin θ , then (1) log x = iθ, log x − = − iθ (2) x n = cis( nθ ) , x − n = cis( − nθ ) ( De Moivre )(3) x n + x − n = 2 cos nθ, x n − x − n = 2 i sin nθ (4) (1 + x ) n = x n/ (cid:18) θ (cid:19) n , (cid:0) x − (cid:1) n = x − n/ (cid:18) θ (cid:19) n (5) (1 − x ) n = e − inπ/ x n/ (cid:18) θ (cid:19) n , (1 − x − ) n = e inπ/ x − n/ (cid:18) θ (cid:19) n (6) e − inπ/ x m + e inπ/ x − m = 2 cos (cid:16) mθ − n π (cid:17) (7) e − inπ/ x m − e inπ/ x − m = 2 i sin (cid:16) mθ − n π (cid:17) First logarithmic function If f ( x ) = x m (1 + x ) n log(1 + x ) where n is a positive integer, then for x = e iθ , f ( x ) = (cid:18) θ (cid:19) n x m + n/ log (cid:18) x / θ (cid:19) = n X k =1 A n,k x k + m + n ! ∞ X k =1 ( − k − ( k − n + k )! x m + n + k with the A n,k defined above in lemma 3.1. Separating the real and imaginary parts yields twoidentities (I,II) from ∞ X k =1 ( − k − ( k − n !( n + k )! cis ( m + n + k ) θ = − n X k =1 A n,k cis ( k + m ) θ + (cid:18) θ (cid:19) n (cid:26) log (cid:18) θ (cid:19) cis (cid:16) m + n (cid:17) θ + i θ (cid:16) m + n (cid:17) θ (cid:27) . The domain of validity is given as − π < θ < π .6. Second logarithmic function If f ( x ) = x m (1 − x ) n log(1 − x ) where n is a positive integer, then two other identities (III,IV)are derived by the method used in the previous section from( − n +1 ∞ X k =1 ( k − n !( n + k )! cis ( m + n + k ) θ = n X k =1 A n,k cis ( k + m ) θ + (cid:18) θ (cid:19) n (cid:26) log(2 sin θ (cid:16) n π − (cid:16) m + n (cid:17) θ (cid:17) + i θ (cid:16) n π − (cid:16) m + n (cid:17) θ (cid:17)(cid:27) The domain of validity is given as − π < θ < π . LISSARD’S TRIGONOMETRIC SERIES 5 Special cases
The four trigonometric series with general term cos( nθ ) /n, sin( nθ ) /n and their alternating ver-sion, (A, B, C, D) in the table above, are obtained by making m → , n → − π < θ < π are next obtained by substituting m = − n in (I,II), or by separating the real and imaginary parts in ∞ X k =1 ( − k − ( k − n !( n + k )! cis kθ = − n X k =1 A n,k cis ( k − n ) θ + (cid:18) θ (cid:19) n (cid:26) log (cid:18) θ (cid:19) cis (cid:16) − n (cid:17) θ + i θ (cid:16) − n (cid:17) θ (cid:27) . Substituting θ = 0 in I’ yields the sum of a numeric alternating series, ∞ X k =1 ( − k − ( k − n !( n + k )! = 2 n log 2 − n X k =1 A k,n . Another numeric alternating series is summed by substituting θ = π/ ∞ X k =1 ( − k − (2 k − n !( n + 2 k − n X k =1 A n,k sin (cid:18) ( n − k ) π (cid:19) + 2 n/ (cid:26) π (cid:16) nπ (cid:17) − sin (cid:16) nπ (cid:17) log (cid:18) θ (cid:19)(cid:27) . Other functions
With f ( x ) = arctan( x ), Blissard sums four series having general term cos[(2 n + 1) θ ] /n , sin[(2 n + 1) θ ] /n and their alternating version, obtaining directly (G, H) then (E,F) with π/ − θ in the table above.The function f ( x ) = arctan(1 / [ x m (1 − x ) n ]) is used to sum two alternating series, ∞ X k =0 ( − k k + 1 cos(2 k + 1) mθ (2 cos θ ) (2 k +1) n = 12 arctan (cid:26) θ ) m cos mθ (2 cos θ ) n − (cid:27) , ∞ X k =0 ( − k k + 1 sin(2 k + 1) mθ (2 cos θ ) (2 k +1) n = 14 log 1 + 2(2 cos θ ) n sin mθ + (2 cos θ ) n − θ ) n sin mθ + (2 cos θ ) n . The function f ( x ) = log(1 + x + x ) generates four new series, after replacing θ by 2 θ : ∞ X k =1 ( − k − k cos 3 kθ (2 cos θ ) k = log (cid:18) θ θ (cid:19) ∞ X k =1 ( − k − k sin 3 kθ (2 cos θ ) k = θ ∞ X k =1 ( − k − k (2 cos θ ) k cos 3 kθ = log (1 + 2 cos 2 θ ) ∞ X k =1 ( − k − k (2 cos θ ) k cos 3 kθ = 2 θ. Next, f ( x ) = 1 + x + . . . + x n + r yields the sum of two other alternating series, ∞ X k =0 ( − k k (cid:18) sin nθ sin rθ (cid:19) k cos ( k ( n + r ) θ ) = log (cid:26) sin( n + r ) θ sin rθ (cid:27) , ∞ X k =0 ( − k k (cid:18) sin nθ sin rθ (cid:19) k sin ( k ( n + r ) θ ) = nθ. JACQUES G´ELINAS
Finally, Blissard derives with f ( x ) = cos x and f ( x ) = sin x two “elegant formulae” :tan(cos θ ) = ∞ X n =0 ( − n cos(2 n + 1) θ (2 n + 1)! ∞ X n =0 ( − n cos(2 n ) θ (2 n )! = ∞ X n =1 ( − n − sin(2 n ) θ (2 n )! ∞ X n =0 ( − n sin(2 n + 1) θ (2 n + 1)! , adding that “these equations hold quite generally”. Indeed, many readers of the Education Times[7, p. 64] noted that, with x = e iθ , the middle quotient can be reduced by using De Moivre’s formulafrom lemma 4.1, tosin x + sin x − cos x + cos x − = 2 sin x + x − cos x − x − x + x − cos x − x − = tan x + x − θ ) . The other “elegant” identity follows likewise fromcos x − − cos x sin x − sin x − = 2 sin x + x − sin x − x − x + x − sin x − x − = tan x + x − θ ) . Heuristic rule for the domain of validity
The author proposes a rule giving the domain of the independent variable for which the closed-form sum formulas remain valid, but without proof, illustrating it with three examples of erroneousresults. If such an infinite trigonometrical series as ∞ X k =0 ( ± k cos m ( p + kr ) θ ( p + kr ) n is capable of being summed in terms of the arc θ , then(1) The range of application is | θ | < π/mr if all signs are positive(2) The range of application is | θ | < π/mr if the signs alternate(3) The endpoints are ordinarily included within the range of application.Blissard adopted later a different point of view, accepting that the sum “is not represented by asingle analytical expression” [25, § π have several ranges of application and a distinct summationcorresponding to each range.” [6]An ingenious method for discovering these “ranges” is to use elementary decompositions (such as4 sin θ = 3 sin θ − sin 3 θ ) and differentiation to get simpler series (having terms such as cos(2 n + 1) θ )which diverge for very obvious values of the variable θ . The formulas for the original series can thenbe recovered by integration. This method is explained further in the section “Discovering thediscontinuities in the sum of a trigonometrical series” of [10] where it is attributed to Stokes (1847)for the case of Fourier series. A simple example from [6] is ∞ X n =0 ( − n cos(2 n + 1) θ (2 n + 1) = π − πθ ≤ θ ≤ π π − π θ + πθ π ≤ θ ≤ π − π + π θ − πθ π ≤ θ ≤ π. “An infinite trigonometrical series is said to be summed when its value is expressed in finite terms” [6, p. 50] LISSARD’S TRIGONOMETRIC SERIES 7
More examples
Blissard stated “We can obtain by the above method numerous trigonometrical formulae, someof which appear to be remarkable. I subjoin some examples which the young student, for whom thispaper is chiefly intended, may work out for himself”. Indeed, it is immediate to find some suitableMaclaurin series, for example in [22, p. 112].log(1 + x )1 + x = ∞ X n =1 ( − n − H n x n , ( | x | < , H n = 1 + 12 + . . . + 1 n )log (cid:16) x sin x (cid:17) = ∞ X n =1 n | B n | n x n (2 n )! , ( | x | < π, B n = Bernoulli number = 1 , , − , , ... )log sec x = ∞ X n =1 n (cid:0) n − (cid:1) | B n | n x n (2 n )! , ( | x | < π , B n = Bernoulli number)The first series corresponds to m = 0 and n = − − π/ < θ < π/ ∞ X n =1 ( − n − H n cis nθ = log (2 cos θ ) + θ tan θ + i [ θ − tan θ log (2 cos θ )] . Numerical verification
The typography in 1860’s England was not totally reliable, in particular in the first volumes ofmathematical publications – there are many noticable errors in the Quaterly Journal, for example.A computer algebra system (CAS) can be used to detect easily missing terms, digit inversions, signerrors, or other obvious misprints. We provide in this section a minimal set of instructions for thefreely available GP/PARI Calculator , from Karim Belabas, Henri Cohen and the PARI Group [1],but other software such as Maple, Mathematica, Python or even Matlab could be chosen instead.In order to detect possible errors in some of the equations of Blissard, we simply verify for somevalues of the parameters ( n = 0 , , . . . , N ) that the left-hand side of the identity is close, in somesense, to the right-hand side. N = 8;DS = 16; \\ number of terms in a Taylor series from \psdefault(seriesprecision,DS);cis(t) = exp(I*t);\\ Floating-point equality (at r=7/8 of precision)near(x, y, r=7/8) = if(x==y, 1, exponent(normlp(Pol(x-y)))/exponent(0.) > r);CHECK(e, msg="") = print( if(!e, "Failed: ", "Passed: "), msg );CHECKN(fn, vn="(n,m)", r=3/4) = CHECK(N+1==sum(n=0,N, near(eval(Str("L",fn,vn)),\eval(Str("R",fn,vn)), r) ), Str(fn) );\\ Binomial summation formula (page 124)L124(n,m) = sum(k=0,n, (-1)^k*binomial(n,k)/(m+k) );R124(n,m) = n! / prod(k=0,n,m+k);CHECKN( 124 ); \\ Induction proof of BlissardLA124(n,m) = L124(n+1,m);RA124(n,m) = L124(n,m) - L124(n,m+1);CHECKN( A124 );LB124(n,m) = R124(n,m) - R124(n,m+1);RB124(n,m) = R124(n+1,m);CHECKN( B124 ); A browser-based implementation is made available at http://pari.math.u-bordeaux.fr/gp.html
JACQUES G´ELINAS \\ Maclaurin logarithm expansionAkn(k,n) = sum(j=1,k, (-1)^(k-j)*binomial(n,j-1)/(k+1-j) );L125(n,x)= sum(k=1,n, Akn(k,n)*x^k ) \+ n!*sum(k=1,DS, (-1)^(k-1)*(k-1)!/(n+k)!*x^(n+k));R125(n,x)= (1+x)^n*log(1+x);CHECKN( 125, "(n,x)" );\\ First logarithmic functionL127(n,m,x) = x^m*L125(n,x);R127(n,m,x) = x^m*R125(n,x);for(m=0,N, CHECKN( 127, Str("(n,",m,",x)") ));\\ Second logarithmic functionL128(n,m,x) = x^m*L125(n,-x);R128(n,m,x) = x^m*R125(n,-x);for(m=0,N, CHECKN( 128, Str("(n,",m,",x)") ));L_127(m,n,t,K=400) = sum(k=1,K,(-1)^(k-1)*n!*(k-1)!/(n+k)!*cis((m+n+k)*t));R_127(m,n,t) = - sum(k=1,n, Akn(k,n)*cis((k+m)*t)) \+ (2*cos(t/2))^n*( log(2*cos(t/2))*cis((m+n/2)*t) + I*t/2*cis((m+n/2)*t) );\\ \sum_{k=1}^\infty (-1)^{k-1} \fr (k-1)!n!/{(n+k)!} = 2^n\log 2 - \sum_{k=1}^n A_{n,k}L130(n,K=400) = sum(k=1,K, 1.0 * (-1)^(k-1) * (k-1)!*n!/(n+k)! );R130(n) = 2^n*log(2) - sum(k=1,n,Akn(k,n));for(n=4,N, CHECK( near(L130(n,400), R130(n), 1/8),\Str("0!/(n+1)!-1!/(n+2)!..., n = ",n) ) );for(n=0,N, CHECK( near(L_127(-n,n,0,400), L130(n,400), 1/8),\Str("L_127(-n,n,0) = L130(n), n = ",n) ) );\\ \sum_{k=1}^\infty (-1)^{k-1} \fr (2k)!n!/{(n+2k+1)!}LM1_131(n,K=400) = sum(k=1,K,(-1)^(k) * 1.0 * (2*k-2)!*n!/(n+2*k-1)!);for(n=2,N, CHECK( near(-imag(L_127(-n,n,Pi/2,400)), LM1_131(n,400), 1/8),\Str("L_127(-n,n,Pi/2) = LM1_131(n), n = ",n) ) );for(n=2,N, CHECK( near(-imag(R_127(-n,n,Pi/2)), LM1_131(n,400), 1/8),\Str("R_127(-n,n,Pi/2) = LM1_131(n), n = ",n) ) );H(n) = sum(k=1,n,1/k); \\ More examplesCHECK( log(1+x)/(1+x) == sum(n=1,DS,(-1)^(n-1)*H(n)*x^n),"log(1+x)/(1+x)");CHECK( log(x/sin(x) ) == sum(n=1,DS/2, m=2*n;\2^m*abs(bernfrac(m))/m*x^m/m!), "log(x/sin x)" );CHECK( log(1/cos(x) ) == sum(n=1,DS/2, m=2*n;\2^m*abs(bernfrac(m))/m*x^m/m!*(2^m-1)), "log(sec x)" );
Acknowledgement
The author thanks Mr Garry Herrington for reviewing an earlier version of this document andcommunicating useful comments, corrections and suggestions to improve it.
LISSARD’S TRIGONOMETRIC SERIES 9
References [1] Karim Belabas, Henri Cohen, and PARI group.
PARI/GP, Computer Algebra System . The Pari Group, Bor-deaux, 1985-2020.[2] John Blissard. Theory of generic equations.
Quart. J. Pure Appl. Math. , 4:279–305, 1861.[3] John Blissard. On the generalization of certain trigonometrical formulae.
Messenger of Mathematics , 1:124–136,1862.[4] John Blissard. Examples of the use and application of representative notation.
Quart. J. Pure Appl. Math. ,6:49–64, 1864.[5] John Blissard. On the generalization of certain formulas investigated by Mr. Walton.
Quart. J. Pure Appl. Math. ,6:167–179, 1864.[6] John Blissard. On the limits and different ranges of application belonging to the summation of infinite trigono-metrical series.
Messenger of Mathematics , 2:50–63, 1864.[7] John Blissard. Question 2114 proposed by Rev. J. Blissard.
Mathematical questions with their solutions fromthe Educational Times , p. 64, 1866. (Edited by W.J. Miller).[8] George Boole.
A treatise on the calculus of finite differences . Chelsea, New York, 1957.[9] Thomas John I’Anson Bromwich.
An introduction to the theory of infinite series . Macmillan, London, 1908.[10] Thomas John I’Anson Bromwich.
An introduction to the theory of infinite series . Macmillan, London, 1926.(second edition revised with the assistance of T.M. Macrobert).[11] Yurii Alexandrovich Brychkov, Oleg Igorevich Marichev, and Anatolii Platonovich Prudnikov.
Integrals andseries I - Elementary functions . Overseas Publishers Association, Amsterdam, 1998. Fourth printing. Translatedfrom the Russian by N.M. Queen.[12] H.S. Carslaw.
Introduction to the theory of Fourier series and integrals . Macmillan, London, 1921. 2nd edition.[13] Costas J. Efthimiou. Trigonometric series via Laplace transforms.
Mathematics Magazine , 79:376–379, 2006(5 pages, arxiv.org/abs/0707.3590 ).[14] Leonard Euler. De eximio usu methodi interpolationum in serierum doctrina.
Opuscula Analytica , 1:157–210,1783. (Opera Omnia: Series 1, Volume 15, 435–497. E555, read in 1772).[15] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.
Concrete mathematics . Addison-Wesley, New York,1989.[16] Eldon R. Hansen.
A table of series and products . Prentice Hall, Englewood Cliffs, New Jersey, 1975.[17] G.H. Hardy.
Divergent series . Oxford University Press, Oxford, 1949.[18] L.B.W. Jolley.
Summation of series . Dover, New York, 1961.[19] Konrad Knopp.
Theory and application of infinite series . Dover, New York, 1954. (Translation by R. C. H.Young of the fourth German edition of 1947).[20] Alexis Marin. Le capo du troupeau des s´eries de Fourier. (6 pages, arxiv.org/abs/1211.3914 ), 2012.[21] Isaac Joachim Schwatt.
An introduction to the operations with series . University of Pennsylvania, Philadelphia,1924.[22] Murray R. Spiegel.
Schaum’s Outline of Mathematical Handbook of Formulas and Tables . McGraw Hill, NewYork, 1968. 1st edition.[23] Georgi P. Tolstov.
Fourier series . Prentice Hall, New Jersey, 1962. (Translated from the Russian by Richard A.Silverman).[24] William Walton. On the transformation of a certain series.
Quart. J. Pure Appl. Math. , 6:22–23, 1864.[25] Edmond Taylor Whittaker and George Neville Watson.
A course of modern analysis . Cambridge UniversityPress, Cambridge, 1927.
Ottawa, Canada
E-mail address ::