How Euler Did It by Ed Sifer A Series of Trigonometric Powers June 008 The story of Euler complex numbers is a complicated one. Earlier in his career, Euler was a champion of equal rights for complex numbers, treating them just like real numbers whenever he could. For example, he showed how to integrate x + dx without using inverse trigonometric functions. He factored x += ( x + ) ( x ), then used partial fractions to rewrite then integrated this difference to get x + = x + x, x + dx = ln x + x. Euler typically omitted constants of integration until he needed them, also seldom used i in place of. His role in that particular notational innovation is exaggerated. Euler struck a second, better-known blow for justice for complex numbers when he took the variable in the exponential function e x to be an imaginary number, say x = θ e θ = cosθ + sinθ., showed that Euler continued to use complex numbers late in his life, but his applications seem to me to be less sweeping more technical, showing how they solved a variety of specific problems. This month we look at one such problem from 773. The title of E447 is "Summatio progressionum sinϕ λ + sinϕ λ + sin3ϕ λ +K + sinnϕ λ, cosϕ λ + cosϕ λ + cos3ϕ λ +K + cosnϕ λ." Right off, this is confusing to the modern reader, because
Euler writes sinϕ λ where we would write sin λ ϕ mean ( sinϕ ) λ. For this, we will use the modern notation. Euler begins by asking us to let cosϕ + sinϕ = p cosϕ sinϕ = q. Then, from de Moivre's formula, we have cosnϕ = pn + q n sinnϕ = pn q n, because sin ϕ + cos ϕ =, we have pq =. Properties of geometric series tell us p α + p α + p 3α +L + p nα = pα q α + q α + q 3α +L q nα = qα ( p nα ) p α ( q nα ) q α. If we add these together repeatedly apply the identities pq = p kα + q kα = coskαϕ (a consequence of de Moivre's formula), we get + cosnαϕ cos ( n + )αϕ. cosαϕ Likewise, if we subtract the q-series from the p-series we get sinαϕ sin( n + )αϕ + sin nαϕ cosαϕ. Euler uses an integral sign,?, where we would use a summation sign,?, so he writes these results as () cosnαϕ cos n + ( p nα + q nα )= + αϕ cosαϕ αϕ sinαϕ + sin nαϕ sin n + ( p nα q nα )= cosαϕ. Now Euler is ready to work on the sums in the title of the article. He takes λ =, his two series become
s = sinϕ + sinϕ + sin 3ϕ +L + sinnϕ = sinnϕ t = cosϕ + cosϕ + cos 3ϕ +L + cosnϕ = cosnϕ. Because of de Moivre's identities, these two series can be rewritten as sinnϕ = pn q n cosnϕ = pn + q n, s = p n q n t = ( p n + q n ). But from formula () above, taking α =, this gives sinϕ + sinnϕ sin n + s = ϕ ( cosϕ) t = cosnϕ cos + ( n + )ϕ. cosϕ Note how unexpectedly simple these formulas are. They each the sum of n terms using only the terms at the beginning the terms at the end, without using any of the terms in between. Now take λ = so that s = sin ϕ + sin ϕ +L + sin nϕ = sin nϕ t = cos ϕ + cos ϕ +L + cos nϕ = cos nϕ. Recalling that pq = we get 3
Similarly, sin nϕ = ( sinnϕ) = pn q n = pn p n q n + q n 4 = pn + q n. 4 cos nϕ = + pn + q n. 4 Summing these, we get 4s = 4t = + ( p n + q n ) ( p n + q n ). Obviously, = n, so, using formula () we get s = n + cosnϕ cos ( n + )ϕ 4 4( cosϕ) t = n cosnϕ cos + ( n + )ϕ. 4 4 cosϕ It is reassuring to note that s + t = n, as it should be, because s is a sum of n squared sines t is a sum of the corresponding squared cosines. Euler does λ = 3 λ = 4, his expressions for s t grow first to three, then to four terms, though the terms grow no more complicated, except for involving higher powers of. Moreover, his expressions have the same general form. Let's look a bit more closely at this expression for s in the case λ =, the sum of the squares of a sequence of sines. Note how the last term does not increase as the number n increases. Also, if cos ϕ is not very close to, then the denominator in the last term is not very small. Moreover, the two cosines in the numerator are always between -, so their difference is between -. Consequently, the last term is bounded between two values, + M M, that do not depend on n. Hence, s is always between n + 4 + M n + M. Thus, as n goes to infinity, so also does s. The same reasoning 4 applies to t. Note that this was not the case for the series corresponding to λ =. The last terms in the expressions for s t are both bounded, by the same argument we gave above, but neither expression contains a term that goes to infinity as n increases. 4
Indeed, these remarks about λ = are true for all odd exponents. That is to say, if λ is odd, then neither s nor t increase without bound as n increases, but for λ even, the series behaves like λ =, both s t grow without bounds. Euler notices this, too, wants to examine it a bit. In Euler's time, there were several notions of the value of a series. One of them, proposed by Jakob Bernoulli, was that the value was the limit of the average value of the partial sums. Using this notion, the series + + + etc. would have value equal to ½, because half the time the partial sums are half the time they are zero. Hence, the weighted average value of the partial sums is ½. This is apparently the notion that justifies Euler's next steps. Taking λ =, he has shown that s = sinϕ + sinϕ + sin 3ϕ +L + sinnϕ sinϕ + sinnϕ sin n + = ϕ. cosϕ Euler argues that the average value of sinnϕ sin( n +)ϕ is zero, so, if we let n go to infinity the value of the now-infinite series s can be considered to be The same analysis makes the infinite series s = sinϕ ( cosϕ). t =. Modern analysts throughout the world cringe at this, because Euler has given an exact, finite sum to two series for which the terms do not converge to zero. The analysts don't let us do that anymore. Perhaps Euler realizes we may have doubts about this particular result, for he reassures us that it is easy to show that this makes sense. He rewrites t as t = cosϕ ( cosϕ). Multiplying both sides by cosϕ writing t as the series it represents, Euler gets cosϕ = ( cosϕ)t ( cosϕ + cos ϕ + cos 3ϕ +L ) = cosϕ = cosϕ + cosϕ + cos 3ϕ + cos4ϕ + etc. cos ϕ cosϕ cosϕ cosϕ cos3ϕ etc. Now, from the angle addition formula for cosines we know that in teneral 5
cosacosb = cos( a b)cos( a + b). Applied to the negative terms of the preceding series, this makes cos ϕ = + cosϕ, cosϕ cos ϕ = cosϕ + cos 3ϕ, cosϕ cos 3ϕ = cosϕ + cos4ϕ, cosϕ cos4ϕ = cos3ϕ + cos5ϕ, cosϕ cos5ϕ = cos4ϕ + cos6ϕ, cosϕ cos6ϕ = cos5ϕ cos 7ϕ, etc. Now, substituting these for those negative terms,, at the same time rearranging the terms a bit, we get cosϕ = cosϕ + cosϕ + cos3ϕ + cos 4ϕ + etc. cosϕ cos ϕ cos 3ϕ cos 4ϕ etc. cos ϕ cos 3ϕ cos 4ϕ etc. Note how, when we substituted + cosϕ for cos ϕ, we put the in the second row of the new expression, the cos ϕ in the third row. Likewise for all the other substitutions. As modern mathematicians, we benefit from the work of Cauchy we know that such rearrangements of terms may not be valid unless the series involved are absolutely convergent, that the series in question here are not absolutely convergent. Today, Euler would have to find another way to do this. Getting back to our formulas, let's rewrite the preceding formula, aligned a bit differently, so that things that cancel can be seen more clearly. We get cosϕ = cosϕ + cosϕ + cos 3ϕ + cos 4ϕ + etc. cosϕ cos ϕ cos 3ϕ cos 4ϕ etc. cos ϕ cos 3ϕ cos 4ϕ etc. which is clearly true. This justifies Euler's claim that, for infinite values of n, t =. Euler thought he was finished, but the Editor's summary at the beginning of the volume of the Novi commentarii mentions that he later added an appendix "Summatio generalis infinitarum aliarum 6
progressionum ad hoc genus referendarum" (Summation of infinitely many general progressions related to this kind). It contains a theorem two examples. Theorem: If we know the sum of a progression then it always permits us to sum the two progressions Az + Bz + Cz 3 + Dz 4 +L + Nz n, S = Axsinϕ + Bx sinϕ + Cx 3 sin3ϕ +L + Nx n sinnϕ T = Axcosϕ + Bx cosϕ + Cx 3 cos3ϕ +L + Nx n cosnϕ. The proof is straightforward, but in the course of the proof, Euler introduces the function notation as he generally uses it in the 760s. When he writes : z, he means us to substitute the function defined by the progression Az + Bz + Cz 3 + Dz 4 +L + Nz n. Though he had used the modern f ( x) function notation briefly in the 730s, Euler did not stick with that notation, from the 760s until his death in 783, he his assistants used this notation with a symbol, usually an upper-case Greek letter, followed by a colon, then the variable. Then he notes that, with p q as before, that is, p = cosϕ + sinϕ q = cosϕ sinϕ, his function notation gives S = : px qx T = : px + : qx. Then he gives examples. Example : If all the coefficients in?:z are equal to if the series is taken to be an infinite series, then : z = z z. Then, from the equations in the proof of his theorem as well as the identities p q = sinϕ, p + q = cosϕ pq =, Euler gets that 7
S = T = xsinϕ x cosϕ + x x cosϕ x x cosϕ + x. Typically, Euler checks that his result agrees with what he already knows. In a corollary, he finds that for the special case x =, this gives back the formulas from earlier in the paper, that sinϕ S = cosϕ T =. As a second example, Euler takes : z = ln z = z + z + 3 z3 + 4 z 4 +L, he finds that xsinϕ S = arctan xcosϕ We leave the details to the reader. T = ln( x cosϕ + x). References: [E447] Euler, Leonhard, Summatio progressionum sinϕ λ + sinϕ λ + sin 3ϕ λ +L + sinnϕ λ, cosϕ λ + cos ϕ λ + cos 3ϕ λ +L + cos nϕ λ, Novi commentarii academiae scientiarum imperialis Petropolitanae, 8 (773) 774, pp. 8-, 4-36. Reprinted in Opera omnia I.5, pp. 68-84. Available online at EulerArchive.org Ed Sifer (SiferE@wcsu.edu) is Professor of Mathematics at Western Connecticut State University in Danbury, CT. He is an avid marathon runner, with 36 Boston Marathons on his shoes, he is Secretary of The Euler Society (www.eulersociety.org). His first book, The Early Mathematics of Leonhard Euler, was published by the MAA in December 006, as part of the celebrations of Euler s tercentennial in 007. The MAA published a collection of forty How Euler Did It columns in June 007. How Euler Did It is updated each month. Copyright 008 Ed Sifer 8