§3. Infinite products
397
has been - a lot more of this kind has been done 29 -, but the machine which
will be capable of proving Dirichlet's Theorem is probably not for tomorrow.
18 - The infinite product for the sine function
Let us start from the identity
(18.1)
xn - yn = (x - y)(x - wy) ... (x _ wn-iy)
where w
exp(2niln), cf. (14.33). On replacing x and y by exp(z) and
exp( - z), the general term of the right hand side becomes
exp(z) - exp( -z + 2kin In) = exp(kin In) [exp(z - kin In) - exp( -z + kin In)].
Since 1 + 2 + ... + n - 1 = n( n - 1) 12, the product of the exponential factors
is equal to exp[(n -1)in/2] = exp(in/2)n-i = i n - i , whence
(18.2)
exp(nz) - exp( -nz) =
= i n - i II[exp(z - kin In) - exp( -z + kinin)].
On replacing z by iz, using the formula
2i. sin z = e iz _ e- iz ,
and regrouping the factors -2i which appear everywhere on the right hand
side, we find
29 Since 1946 we have seen number theorist specialists in the USA use the first
electronic calculator, the ENIAC of Eckert and Mauchly, for arithmetic calculations; these clearly did not interest the "serious" users, but enabled them to
check the functioning of the machine and to invent much more useful methods
of calculation, as related by Herman H. Goldstine, The Computer from Pascal to
von Neumann (Princeton UP, 1972), particularly pp. 233 and 273 Ii propos D. H.
Lehmer; in particular they calculated 2,000 decimal places of 7r. When the first
programmable computer constructed by John von Neumann (there were also
Eckert and Mauchly and their UNIVAC of 1950) at the Institute for Advanced
Study at Princeton between 1945 and 1952 became operational, the exploit was
celebrated in the course of an inauguration ceremony, where they exhibited calculations performed for Emil Artin, one of the "fathers" of "abstract" algebra,
to show that a famous arithmetic conjecture of Kummer's, a century old, was
probably incorrect. For the course of the story it may be better to rely on von
Neumann himself: "As far as the Institute is concerned, and the people who were
there are concerned, this computer came into operation in 1952, after which the
first large problem that was done on it, and which was quite large and took even
under these conditions half a year, was for the thermonuclear problem. Previous
to that I had spent a lot of time on calculations on other computers for the thermonuclear problem." (In the Matter of J. Robert Oppenheimer, United States
Atomic Energy Commission, 1954, reprint MIT Press, 1971, p. 655, testimony
of von Neumann to the Oppenheimer "trial".) Not mentioned in Goldstine.
397
has been - a lot more of this kind has been done 29 -, but the machine which
will be capable of proving Dirichlet's Theorem is probably not for tomorrow.
18 - The infinite product for the sine function
Let us start from the identity
(18.1)
xn - yn = (x - y)(x - wy) ... (x _ wn-iy)
where w
exp(2niln), cf. (14.33). On replacing x and y by exp(z) and
exp( - z), the general term of the right hand side becomes
exp(z) - exp( -z + 2kin In) = exp(kin In) [exp(z - kin In) - exp( -z + kin In)].
Since 1 + 2 + ... + n - 1 = n( n - 1) 12, the product of the exponential factors
is equal to exp[(n -1)in/2] = exp(in/2)n-i = i n - i , whence
(18.2)
exp(nz) - exp( -nz) =
= i n - i II[exp(z - kin In) - exp( -z + kinin)].
On replacing z by iz, using the formula
2i. sin z = e iz _ e- iz ,
and regrouping the factors -2i which appear everywhere on the right hand
side, we find
29 Since 1946 we have seen number theorist specialists in the USA use the first
electronic calculator, the ENIAC of Eckert and Mauchly, for arithmetic calculations; these clearly did not interest the "serious" users, but enabled them to
check the functioning of the machine and to invent much more useful methods
of calculation, as related by Herman H. Goldstine, The Computer from Pascal to
von Neumann (Princeton UP, 1972), particularly pp. 233 and 273 Ii propos D. H.
Lehmer; in particular they calculated 2,000 decimal places of 7r. When the first
programmable computer constructed by John von Neumann (there were also
Eckert and Mauchly and their UNIVAC of 1950) at the Institute for Advanced
Study at Princeton between 1945 and 1952 became operational, the exploit was
celebrated in the course of an inauguration ceremony, where they exhibited calculations performed for Emil Artin, one of the "fathers" of "abstract" algebra,
to show that a famous arithmetic conjecture of Kummer's, a century old, was
probably incorrect. For the course of the story it may be better to rely on von
Neumann himself: "As far as the Institute is concerned, and the people who were
there are concerned, this computer came into operation in 1952, after which the
first large problem that was done on it, and which was quite large and took even
under these conditions half a year, was for the thermonuclear problem. Previous
to that I had spent a lot of time on calculations on other computers for the thermonuclear problem." (In the Matter of J. Robert Oppenheimer, United States
Atomic Energy Commission, 1954, reprint MIT Press, 1971, p. 655, testimony
of von Neumann to the Oppenheimer "trial".) Not mentioned in Goldstine.