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Equation Domain Method ∞ ∞ n cos nθ θ reiθ A = − log 2 sin 0 <θ< 2π , |r|≤ 1 n  2 n
n=1 n=1 X∞ X sin nθ π − θ B = 0 <θ< 2π [21, p. 211] n 2 n=1 X∞ − cos nθ θ C (−1)n 1 = log 2cos −π<θ<π θ → π − θ in A n  2 n=1 X∞ − sin nθ θ D (−1)n 1 = −π<θ<π θ → π − θ in B n 2 n=1 X∞ cos(2n + 1)θ 1 θ E = − log tan 0 <θ<π A+C 2n +1 2  2 n=0 X∞ sin(2n + 1)θ π F = 0 <θ<π B + D 2n +1 4 n=0 X∞ cos(2n + 1)θ π π π π G (−1)n = − <θ< θ → − θ in F 2n +1 4 2 2 2 n=0 X∞ sin(2n + 1)θ 1 1 + sin θ π π π H (−1)n = log − <θ< θ → − θ in E 2n +1 4 1 − sin θ 2 2 2 n=0 X∞ cos2nπθ (−1)k−1 B (θ) I = 2k 0 ≤ θ ≤ 1; k =1, 2,... [9, p. 256], [10, p. 370] (2nπ)2k 2 (2k)! nX=1 ∞ sin 2nπθ (−1)k−1 B (θ) 0 <θ< 1; k =0 J = 2k+1 [9, p. 256], [10, p. 370] (2nπ)2k+1 2 (2k + 1)! nX=1 0 ≤ θ ≤ 1; k =1, 2,... ∞ cos(n − a)θ π K = − 0 <θ< 2π; a ∈ R [9, p. 230], [10, p. 371] n − a tan(aπ) n=X−∞ ∞ sin(n − a)θ L = π 0 <θ< 2π; a ∈ R [9, p. 230], [10, p. 371] n − a n=X−∞ ∞ cos nθ 7π4 π2θ2 θ4 θ 1 M (−1)n+1 = − + −π ≤ θ ≤ π k =2, θ → + in I n4 720 24 48 2π 2 n=1 X∞ sin2 nθ π2θ2 θ4 π π 1 − cos2nθ N (−1)n+1 = − − ≤ θ ≤ sin2 nθ = n4 12 6 2 2 2 nX=1 BLISSARD’S TRIGONOMETRIC SERIES 3

2. 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 n 1 n! (−1)k = , (m> 0,n =0, 1, 2,...). km + k n (m + k) Xk=0 k=0 Q Proof. With Blissard’s representative notation, let Rk =1/k, and downgrade exponents into indices 1 m n n m+k after expansion . It is thus required to prove R (1 − R) = n! k=0 R , which is an identity if n = 0. If we assume that this has been proven for a certain integerQ n ≥ 0 and all m> 0, then Rm (1 − R)n+1 = Rm (1 − R) (1 − R)n = Rm (1 − R)n − Rm+1 (1 − R)n n n n = n! Rm+k − n! Rm+1+k = Rm − Rm+n+1 n! Rm+k kY=0 kY=0
kY=1 n 1 1 n +1 = (n + 1) Rm Rm+n+1 n! Rm+k − = m m + n +1 m(m + n + 1) kY=1 n+1 = (n + 1)! Rm+k. kY=0 Thus the identity in question holds for the next integer n + 1 and all m> 0. 

Since the same term 1/m appears on both sides, the identity can be said to hold also as m → 0. Blissard used the same induction, but less concise notation with a few explicit terms and “&c.”. He gave a longer proof using polynomials and differentiation in [5, p. 169]. Another simple proof, similar to [24] minus a trigononetric substitution, uses the recurrence relation of the Beta function :

n 1 m 1 1 n 1 − x n − (−1)k = xm 1(1 − x)n dx = (1 − x)n + xm(1 − x)n 1 dx km + k Z m m Z Xk=0 0 0 0 1 n(n − 1) − Γ(n + 1)Γ(m) = xm+1(1 − x)n 2 dx = ... = . m(m + 1) Z0 Γ(m + n + 1) ∞ −mx Boole [8, p. 26] obtains this from the n-th difference of 1/m = 0 e dx, a method of Abel, and one can also use differences of falling factorial powers [15, p. 188]R or partial fraction decomposition.

3. Maclaurin logarithm expansion A second lemma uses the first one to obtain the expansion of a binomial polynomial times a logarithm via the Cauchy product of absolutely convergent power series (we omit the proof). The An,k will appear in the examples presented below. Lemma 3.1. If n is a positive integer and |x| < 1, then

n ∞ − (k − 1)! (1 + x)n log(1 + x)= A xk + n! (−1)k 1 xn+k, n,k (n + k)! kX=1 kX=1 where k − n 1 A := (−1)k j . n,k j − 1k − j +1 Xj=1

1 k m n m+n m n m n R := Rk for k > 0. We use R R = R and R C − R C = (R − R )C for m> 0,n> 0. 4 JACQUES GELINAS´

4. The general recipe A simple method for discovering trigonometric series with a closed-form sum is to start from the Maclaurin expansion of a known function of x and set x = eiθ, as was done above for the geometric series. Blissard states that this works, “whatever form f(x) may assume”, for the expressions f(x) ± f(x−1) which “can be evaluated in terms of trigonometrical functions of θ” with the help of the 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 = eiθ and cis θ := cos θ + i sin θ , then

− (1) log x = iθ, log x 1 = −iθ − (2) xn = cis(nθ), x n = cis(−nθ) (De Moivre) − − (3) xn + x n = 2cos nθ, xn − x n =2i sin nθ n n θ − n − θ (4) (1 + x)n = xn/2 2cos , 1+ x 1 = x n/2 2cos  2  2 n
n − θ − − θ (5) (1 − x)n = e inπ/2xn/2 2 sin , (1 − x 1)n = einπ/2x n/2 2 sin  2  2 − − π (6) e inπ/2xm + einπ/2x m = 2cos mθ − n  2  − − π (7) e inπ/2xm − einπ/2x m =2i sin mθ − n  2 

5. First logarithmic function If f(x)= xm(1 + x)n log(1 + x) where n is a positive integer, then for x = eiθ,

θ n θ f(x)= 2cos xm+n/2 log x1/22cos  2  2 n ∞ − (k − 1)! = A xk+m + n! (−1)k 1 xm+n+k n,k (n + k)! Xk=1 kX=1 with the An,k defined above in lemma 3.1. Separating the real and imaginary parts yields two identities (I,II) from

∞ n − (k − 1)!n! (−1)k 1 cis (m + n + k) θ = − A cis (k + m) θ (n + k)! n,k kX=1 Xk=1 θ n θ n θ n + 2cos log 2cos cis m + θ + i cis m + θ .  2   2  2  2  2   The domain of validity is given as −π<θ<π.

6. Second logarithmic function If f(x) = xm(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 (k − 1)!n! (−1)n+1 cis (m + n + k) θ = A cis (k + m) θ (n + k)! n,k Xk=1 kX=1 θ n θ π n θ π n + 2 sin log(2 sin ) cis n − m + θ + i cis n − m + θ  2  2  2  2   2  2  2   The domain of validity is given as −2π<θ< 2π. BLISSARD’S TRIGONOMETRIC SERIES 5

7. 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 → 0,n → 0 in the four previous identities (I,II,III,IV) and these examples, first announced in [2, p. 281], justify the word “general- ization” in the original title chosen by Blissard. Two identities (I’,II’) with domain of validity −π<θ<π are next obtained by substituting m = −n in (I,II), or by separating the real and imaginary parts in ∞ n − (k − 1)!n! (−1)k 1 cis kθ = − A cis (k − n) θ (n + k)! n,k Xk=1 Xk=1 θ n θ n θ n + 2cos log 2cos cis − θ + i cis − θ .  2   2  2  2  2   Substituting θ = 0 in I’ yields the sum of a numeric alternating series, ∞ n − (k − 1)!n! (−1)k 1 =2n log 2 − A . (n + k)! k,n kX=1 kX=1 Another numeric alternating series is summed by substituting θ = π/2 in II’, ∞ n − (2k − 2)!n! (n − k)π (−1)k 1 = A sin (n +2k − 1)! n,k  2  Xk=1 kX=1 π nπ nπ θ +2n/2 cos − sin log 2cos .  4  4   4   2 8. Other functions With f(x) = arctan(x), Blissard sums four series having general term cos[(2n + 1)θ]/n, sin[(2n + 1)θ]/n and their alternating version, obtaining directly (G, H) then (E,F) with π/2 − θ in the table above. The function f(x) = arctan(1/[xm(1 − x)n]) is used to sum two alternating series, ∞ (−1)k cos(2k + 1)mθ 1 2(2 cos θ)m cos mθ = arctan , 2k +1 (2 cos θ)(2k+1)n 2  (2 cos θ)2n − 1  kX=0 ∞ (−1)k sin(2k + 1)mθ 1 1 + 2(2cos θ)n sin mθ + (2cos θ)2n = log . 2k +1 (2 cos θ)(2k+1)n 4 1 − 2(2 cos θ)n sin mθ + (2cos θ)2n kX=0 The function f(x) = log(1 + x + x2) generates four new series, after replacing θ by 2θ : ∞ (−1)k−1 cos3kθ 1+2cos2θ = log k (2 cos θ)k  2cos θ  Xk=1 ∞ (−1)k−1 sin 3kθ = θ k (2 cos θ)k Xk=1 ∞ (−1)k−1 (2 cos θ)k cos3kθ = log(1+2cos2θ) k kX=1 ∞ (−1)k−1 (2 cos θ)k cos3kθ =2θ. k kX=1 Next, f(x)=1+ x + ... + xn+r yields the sum of two other alternating series, ∞ (−1)k sin nθ k sin(n + r)θ cos(k(n + r)θ) = log , k  sin rθ   sin rθ  Xk=0 ∞ (−1)k sin nθ k sin (k(n + r)θ)= nθ. k  sin rθ  kX=0 6 JACQUES GELINAS´

Finally, Blissard derives with f(x) = cos x and f(x) = sin x two “elegant formulae” :

∞ ∞ cos(2n + 1)θ − sin(2n)θ (−1)n (−1)n 1 (2n + 1)! (2n)! nX=0 nX=1 tan(cos θ)= ∞ = ∞ , cos(2n)θ sin(2n + 1)θ (−1)n (−1)n (2n)! (2n + 1)! nX=0 nX=0 adding that “these equations hold quite generally”. Indeed, many readers of the Education Times [7, p. 64] noted that, with x = eiθ, the middle quotient can be reduced by using De Moivre’s formula from lemma 4.1, to

−1 −1 −1 x+x x−x −1 sin x + sin x 2 sin 2 cos 2 x + x = −1 −1 = tan = tan(cos θ). cos x + cos x−1 x+x x−x 2 2cos 2 cos 2 The other “elegant” identity follows likewise from

−1 −1 −1 x+x x−x −1 cos x − cos x 2 sin 2 sin 2 x + x = −1 −1 = tan = tan(cos θ). sin x − sin x−1 x+x x−x 2 2cos 2 sin 2

9. 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 erroneous results. If such an infinite trigonometrical series as ∞ cosm(p + kr)θ (±1)k (p + kr)n kX=0

is capable of being summed2 in terms of the arc θ, then (1) The range of application is |θ| < 2π/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 a single analytical expression” [25, §9.11] : “Further investigation of this important and interesting subject has however led me to perceive that such series may for their whole period i.e. from all values assigned for the arc from 0 to 2π have several ranges of application and a distinct summation corresponding to each range.” [6] An ingenious method for discovering these “ranges” is to use elementary decompositions (such as 4 sin3 θ = 3 sin θ−sin 3θ) and differentiation to get simpler series (having terms such as cos(2n+1)θ) which diverge for very obvious values of the variable θ. The formulas for the original series can then be recovered by integration. This method is explained further in the section “Discovering the discontinuities 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

3 2 π πθ π ∞ − 0 ≤ θ ≤ 323 82 2 2 n cos(2n + 1)θ  3π π θ πθ π 3π (−1) 3 = 32 − 4 + 8 2 ≤ θ ≤ 2 (2n + 1)  3 2 2 nX=0  −15π π θ πθ 3π 32 + 2 − 8 2 ≤ θ ≤ 2π. 

2“An infinite trigonometrical series is said to be summed when its value is expressed in finite terms” [6, p. 50] BLISSARD’S TRIGONOMETRIC SERIES 7

10. More examples Blissard stated “We can obtain by the above method numerous trigonometrical formulae, some of which appear to be remarkable. I subjoin some examples which the young student, for whom this paper is chiefly intended, may work out for himself”. Indeed, it is immediate to find some suitable Maclaurin series, for example in [22, p. 112]. ∞ log(1 + x) − 1 1 = (−1)n 1H xn, (|x| < 1,H =1+ + ... + ) 1+ x n n 2 n nX=1 ∞ x |B | x2n 1 1 1 log = 22n 2n , (|x| < π,B = Bernoulli number = 1, , − , , ...) sin x 2n (2n)! 2n 6 30 42   nX=1 ∞ |B | x2n π log sec x = 22n 22n − 1 2n , (|x| < ,B = Bernoulli number) 2n (2n)! 2 2n nX=1
The first series corresponds to m = 0 and n = −1 in the first logarithmic function above and yields, for −π/2 <θ<π/2, ∞ n−1 2 (−1) Hn cis nθ = log (2 cos θ)+ θ tan θ + i [θ − tan θ log (2 cos θ)] . nX=1 11. Numerical verification The typography in 1860’s England was not totally reliable, in particular in the first volumes of mathematical 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, sign errors, or other obvious misprints. We provide in this section a minimal set of instructions for the freely available GP/PARI Calculator3, 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 some values of the parameters (n = 0, 1,...,N) that the left-hand side of the identity is close, in some sense, to the right-hand side. N =8; DS = 16; \\ number of terms in a Taylor series from \ps default(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 Blissard LA124(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 );

3A browser-based implementation is made available at http://pari.math.u-bordeaux.fr/gp.html 8 JACQUES GELINAS´

\\ Maclaurin logarithm expansion Akn(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 function L127(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 function L128(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 examples

CHECK( 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)" );

12. Acknowledgement The author thanks Mr Garry Herrington for reviewing an earlier version of this document and communicating useful comments, corrections and suggestions to improve it. BLISSARD’S TRIGONOMETRIC SERIES 9

References

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