Are there really infinite transcendental numbers?

Inverse trigonometric and hyperbolic functions are transcendental for their argument algebraic and nonzero -- a consequence of the Lindemann-Weierstrass theorem

arctan(x)/pi is transcendental for x rational -- a consequence of the Gelfond-Schneider theorem

Transcendentality of zeros of higher dereivatives of functions involving Bessel functions - Lorch - 1995 - International Journal of Mathematics and Mathematical Sciences - Wiley Online Library

Bessel functions J(n,x), their first derivatives J'(n,x) = (d/dx)J(n,x), and one divided by the other, J'(n,x)/J(n,x), are all transcendental for n rational and x algebraic and nonzero.

All nonzero roots of J(n,x) and J'(n,x) are transcendental if n is rational.

 Lemniscate constant - ratio of perimeter to "diameter"
\( (x^2+y^2)^2 = (x^2 - y^2) \)

For polar coordinates, \( x = r \cos\theta ,\ y = r \sin\theta \), where \( r = \sqrt{\cos 2\theta} \)

The distance s along a curve is given by this integral, in both rectangular and polar coordinates, for curve parameter t:

\( \displaystyle{ s = \int \left[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 \right] dt = \int \left[ \left( \frac{dr}{dt} \right)^2 + \left( r \frac{d\theta}{dt} \right)^2 \right] dt } \)

The lemniscate constant, written as an alternative form of "pi":

\( \displaystyle{ 2\varpi = \int_{-\pi/4}^{\pi/4} + \int_{3\pi/4}^{5\pi/4} \frac{1}{\sqrt{\cos 2\theta}} d\theta } \)
giving
\( \displaystyle{ \varpi = 2 \int_0^{\pi/4} \frac{1}{\sqrt{\cos 2\theta}} d\theta = 2 \int_0^{1/\sqrt{2}} \frac{du}{\sqrt{(1-u^2)(1-2u^2)}} = 2 \int_0^1 \frac{dv}{\sqrt{(1-v^2)(2-v^2)}} = 2 \int_0^1 \frac{dw}{\sqrt{1-w^4}} } \)
doing substitutions
\( \theta = \arcsin u ,\ u = v/\sqrt{2} ,\ v = \sqrt{1-w^2} \)

\( \displaystyle{ \varpi = 2 K(-1) = \sqrt{2} K \left(\tfrac12\right) = \tfrac12 \Beta \left( \tfrac14 , \tfrac12 \right) = \frac{ \Gamma \left( \tfrac14 \right)^2}{2 \sqrt{2\pi}} = \frac{2 - \sqrt{2}}{4} \frac{ \zeta \left( \tfrac34 \right)^2}{ \zeta \left( \tfrac14 \right)^2} } \)

It is transcendental, and algebraically independent of pi.

More on periods (the numbers):  Period (algebraic geometry) mentioning M. Kontsevich's and D. Zagier's paper Periods

A real number is a period if it can be expressed as:

integral over (P(X) > 0) of Q(X) dX

where X is some variables, P is a polynomial with rational coefficients, and Q is a rational function with rational coefficients; a rational function is a polynomial divided by a polynomial.

A complex number is a period if its real and imaginary parts are periods.

Periods are closed under addition and multiplication, making them a ring, just like the integers. All algebraic numbers are periods, as are natural logarithms of positive algebraic numbers, pi, and elliptic integrals with rational arguments.

All periods are computable numbers, but there are computable numbers that are not periods, and [0805.0349] Periods and elementary real numbers gives an example.

 Elliptic integral - in general,

\( \displaystyle{ \int R(x,\sqrt{P(x)}) \, dx } \)

where R is a rational function, the ratio of two polynomials, and P is a cubic or quartic polynomial without repeated roots.

Every integral of this form can be reduced to a combination of integrals of rational functions and these three kinds of elliptic integrals:

\( \displaystyle{ \text{ I } \int \frac{dx}{\sqrt{(1-x^2)(1-m x^2)}} \text{ -- II } \int dx \sqrt{ \frac{1 - m x^2}{1 - x^2}} \text { -- III } \int \frac{dx}{(1 - n x^2) \sqrt{(1-x^2)(1-m x^2)}} } \)

These are typically expressed in  Legendre form though there are several notation conventions. The incomplete elliptic integral of the first kind:

\( \displaystyle{ F(x;k) = \int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}} } \)

\(\displaystyle{ F(\varphi,k) = F(\varphi|k^2) = F(\sin\varphi;k) = \int_0^\varphi \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} } \)

\( \displaystyle{ F(\varphi \backslash \alpha) = F(\varphi,\sin\alpha) = \int_0^\varphi \frac{d\theta}{\sqrt{1-\sin^2\alpha \sin^2\theta}} } \)

Here, k is the elliptic modulus or eccentricity, k^2 is the parameter, and arcsin(k) is the modular angle.

The incomplete elliptic integral of the second kind:

\( \displaystyle{ E(\varphi,k) = \int_0^\varphi d\theta \sqrt{1-k^2 \sin^2\theta} } \)

The incomplete elliptic integral of the third kind:

\(\displaystyle{ \Pi(n;\varphi,k) = \int_0^\varphi \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2 \sin^2\theta}} } \)

The complete elliptic integrals:

\(\displaystyle{ K(k) = F(\tfrac{\pi}{2},k) = F(1;k),\ E(k) = E(\tfrac{\pi}{2},k) = E(1;k) ,\ \Pi(n;k) = \Pi(n;\tfrac{\pi}{2},k) = \Pi(n;1;k) } \)

Complementary values are also defined: m' = 1 - m, k' = sqrt(1 - k^2), a' = pi/2 - a
where k = sin(a) and m = k^2.

For some modulus / parameter value,  Landen's transformation relates the elliptic integrals for that value to those of some greater or lesser values, and repeating this transform several times will get the integral into a form that is easy to calculate.

An alternate formulation is  Carlson symmetric form

\( \displaystyle{ R_F(x,y,z) = \frac12 \int_0^\infty \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}} } \)

\( \displaystyle{ R_J(x,y,z,p) = \frac32 \int_0^\infty \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}} } \)

\( \displaystyle{ R_G(x,y,z) = \frac14 \int_0^\infty \frac{t \, dt}{\sqrt{(t+x)(t+y)(t+z)}} \left( \frac{x}{t+x} + \frac{y}{t+y} + \frac{z}{t+z} \right)} \)

\( \displaystyle{ R_C(x,y) = R_F(x,y,y) ,\ R_D(x,y,z) = R_J((x,y,z,z) } \)

From elliptic integrals one can define elliptic functions.

 Jacobi elliptic functions - I'll use a comma for the parameter instead of the modulus. Thus, the incomplete elliptic integral of the first kind is

u = F(p,m) = integral for a from 0 to p for 1/sqrt(1 - m*sin(a)^2)

Inverting, p = am(u,m) -- the Jacobi amplitude

The three basic Jacobi elliptic functions are

• sn(u,m) = sin(am(u,m))
• cn(u,m) = cos(am(u,m))
• dn(u,m) = (d/du) am(u,m) = sqrt(1 - m*sn(u,m)^2)

Note that sn(u,m)^2 + cn(u,m)^2 = 1.

There are twelve of them, and the others can be derived from these ones with identities

pq * p'q' = pq' * p'q

• pr / qr = pq
• pr * rq = pq
• 1 / pq = qp

suppressing the args. One can use  Neville theta functions th(x,u,m)
pq(u,m) = th(p,u,m) / th(q,u,m)

Likewise, one can express these elliptic functions with Jacobi  Theta function Both Neville and Jacobi kinds are Fourier series in the arg u where the coefficient for sin or cos of n*u/(something) is proportional to q^(n^2) or q^(n(n+1)) where q is the  Nome (mathematics) - that makes a super fast converging series unless q is close to 1.

q(m) = exp(-pi*K(m)/K(m')) ~ m/16 + m^2/32 + 21*m^3/1024 + 31*m^4/2048 + ...

q(1/2) = 0.0432139...

q(0.999989522) = 0.5

About Jacobi elliptic functions, they have two periods: a real period 4K(m) and an imaginary period 4i*K(m') That seems odd at first, but that is made possible by complex numbers being decomposable into pairs of real numbers.

These functions have various identities for negation of argument u, multiplying by i, addition and subtraction of quarter periods K(m) and i*K(m'), and of half periods 2*K(m) and 2i*K(m'), and addition and differentiation formulas, much like for trigonometric and hyperbolic functions.

They also have formulas for negating parameter m and taking its reciprocal, much like similar formulas for the elliptic integrals.

Special cases: m = 0 (trigonometric functions) and m = 1 (hyperbolic functions):

sn(u,0) = sin(u), cn(u,0) = cos(u), dn(u,0) = 1
sd(u,0) = sin(u), cd(u,0) = cos(u), nd(u,0) = 1
sc(u,0) = tan(u), nc(u,0) = sec(u), dc(u,0) = sec(u)
ns(u,0) = csc(u), cs(u,0) = cot(u), ds(u,0) = csc(u)

sn(u,1) = tanh(u), cn(u,1) = sech(u), dn(u,1) = sech(u)
sd(u,1) = sinh(u), cd(u,1) = 1, nd(u,1) = cosh(u)
sc(u,1) = sinh(u), nc(u,1) = cosh(u), dc(u,1) = 1
ns(u,1) = coth(u), cs(u,1) = csch(u), ds(u,1) = csch(u)

These are degenerate cases, with one of the two periods infinite.

In general, u -> i*u' does m -> m' = 1 - m

 Weierstrass elliptic function - an alternate formulation, but equivalent to the Jacobi ones.

These are usually written with a fancy script P, but I'll use plain P here.

For parameters g2, g3, and arg z, P(z) satisfies

(P'(z))^2 = 4(P(z))^3 - g2*P(z) - g3

These are related to elliptic curves, with a point on such a curve having coordinates {P(z),P'(z)}. Weierstrass elliptic functions have addition formulas, and these translate into addition formulas for points on elliptic curves. Using instead of complex numbers some algebraic field like Z(big prime) or Z2^(big number), one gets an addition formula that is used in some cryptographic algorithms.

Weierstrass functions can be expressed in terms of Jacobi ones.

 Lemniscate elliptic functions

Defined by differential equations

(d/dz)sl(z) = (1 + sl(z)^2)*cl(z)
(d/dz)cl(z) = - (1 + cl(z)^2)*sl(z)

with identity cl(z)^2 + sl(z)^2 + cl(z)^2*sl(z)^2 = 1 or (1 + cl(z)^2)*(1 + sl(z)^2) = 2

They also can be expressed in terms of Jacobi elliptic functions: sl(z) = sn(z,-1) and cl(z) = cd(z,-1)

AWSLecture3.pdf and contents.dvi - Waldschmidt1.pdf and (duplicate?) contents.dvi - SurveyTrdceEllipt2006.pdf -- Elliptic Functions and Transcendence by Michel Waldschmidt

He starts out by reviewing some transcendence results associated with the exponential function and its inverse, the logarithmic function.

• Hermite 1873: e is transcendental.
• Lindemann 1882: pi is transcendental.
• Hermite-Lindemann 1882: the nonzero logarithm of any nonzero algebraic number is transcendental.
• Hermite-Lindemann 1882: the exponential of any nonzero algebraic number is transcendental.
• Lindemann–Weierstrass 1885: for a1, ..., an algebraic numbers that are linearly independent over the rational numbers, e^a1, ..., e^an are algebraically independent.
• Lindemann–Weierstrass 1885: for a1, ..., an distinct algebraic numbers, e^a1, ..., e^an are linearly independent over the algebraic numbers.
• Gel'fond–Schneider 1934: for a a nonzero algebraic number and b an irrational algebraic number, then a^b = exp(b*log(a)) is transcendental.
• Gel'fond–Schneider 1934: for algebraic numbers a1, a2 with nonzero logarithms, and for log(a1)/log(a2) irrational, then that ratio is transcendental.
• Baker 1966: let log(a1), ..., log(an) be logarithms of algebraic numbers that are linearly independent over the rational numbers. Then 1, log(a1), ..., log(an) are linearly independent over the algebraic numbers.

Then the Six Exponentials Theorem.

Consider n complex numbers x1, ..., xn that are linearly independent in the rational numbers, and presumably all nonzero, and m complex numbers y1, ..., ym also with that property. If m*n > m + n, then at least one of e^(xi*yj) is transcendental.

There is a Four Exponentials Conjecture which states that that is true for m = n = 2 and thus that m*n = m + n. So let us consider m*n >= m + n.

Let us see what (m,n) values are possible. Let n >= m for convenience. Try m = 1 first. n >= n + 1 not possible. So m >= 2. That means n >= m/(m-1). The right-hand side is only an integer if m = 2, and that gives n >= 2. Thus, the only positive-integer solution of m*n = m + n is m = n = 2.

n >= m/(m-1) gives us n >= 1 + 1/(m-1) <= 2. Thus, for m*n > m + n, m = 2 gives us n >= 3 and m and n can both be >= 3.

The six comes from the smallest number that makes at least one transcendental number: n = 3, m = 2, n*m = 6.

The four comes from that conjectured number: n = m = 2, n*m = 4.

For m = 1, I will show that there is a case where all the e^(x*y) are algebraic. Set y = 1 and set the x's to natural logarithms of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19... this means that their natural logarithms are linearly independent over the rational numbers, since none of them is a factor of any other of them. By Baker's theorem, 1 and their logs are linearly independent over the algebraic numbers, meaning that the logs are all transcendental numbers.

But the e^(x*y) = e^(log(p)*1) = p -- algebraic. So there is a case of all-algebraic for m = 1.

Next is  Algebraic independence

Pi by itself is algebraically independent, as is sqrt(pi) by itself, but together, they are not:

pi - (sqrt(pi))^2 = 0

Using the sort of numbers in the six-exponentials theorem and the four-exponentials conjecture,

• If m*n >= 2(m+n) then at least two of the e^(xi*yj) are algebraically independent.
• If m*n >= m + 2n, then at least two of the xi and e^(xi*yj) are algebraically independent (n of xi).
• If m*n > m + n, then at least two of the xi and yj and e^(xi*yj) are algebraically independent (m of yj also).
• If m = n = 2, and if e^(x1*y1) and e^(x2*y1) are algebraic, then at least two of x1, x2, y1, y2, e^(x1*y2), e^(x2*y2) are algebraically independent.

For x1 = y1 = i*pi and x2 = y2 = 1, one finds that at least one of (e^(pi^2) is transcendental) and (e and pi are algebraically independent) is true.

It's theorems like that which give results like proving that only some numbers out of some set are transcendental.

Michel Waldschmidt then mentions D. Rohrlich's conjecture that every multiplicative relationship between values of the Euler gamma function are derived from these three:

Translation: \( \displaystyle{ \Gamma(z+1) = z \Gamma(z) } \)

Reflection: \( \displaystyle{ \Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin (\pi z)} } \)

Multiplication: \( \displaystyle{ \prod_{k=0}^{n-1} \Gamma \left( z + \frac{k}{n} \right) = (2\pi)^{(n-1)/2} n^{-nz+(1/2)} \Gamma(nz) } \)


After discussing elliptic functions and elliptic curves, he gets into transcendence results for elliptic integrals.

Siegel 1932: Consider a Weierstrass elliptic function P(z) with invariants g2, g3, and periods w1, w2. Then if both g2 and g3 are algebraic, then at least one of w1 and w2 is transcendental.

P(z + n1*w1 + n2*w2) = P(z) for integer n1, n2

The text is not very clear, but it seems to imply that both w1 and w2 are transcendental. That means that the complete elliptic integrals K and E are transcendental if the parameter is algebraic.

Before continuing further, I note zeta(z) = - integral of P(z) over z (not the Riemann one!)
It is quasiperiodic: zeta(z + n1*w1 + n2*w2) = zeta(z) + n1*eta1 + n2*eta2

• Schneider 1936: like Siegel 1932, but w1, w2, eta1, eta2 are all transcendental.
• Schneider 1936 (1.1): For Weierstrass elliptic function P with algebraic invariants g2, g3, then if x is algebraic and nonzero, then P(x) is finite and transcendental.
• Schneider 1936 (1.2): For algebraic numbers a, b, at least one nonzero, then for every u that's not on the period lattice of P, at least one is transcendental of P(u) and a*u + b*zeta(u)
• Schneider 1936 (2): For P1 and P2 two algebraically independent Weierstrass elliptic functions with algebraic invariants, then if both P1(x) and P2(x) are finite, then at least one is transcendental.
• Schneider 1936 (3): For x not on the period lattice, at least one of P(x) and e^x is transcendental.

This means that incomplete elliptic integrals F and E are transcendental if their parameters are algebraic, if their integrands are square roots of rational functions without trigonometric functions in them.

Then an elliptic-function of the six-exponentials theorem, which I found hard to follow.

Then algebraic independence. Some of the results are difficult to follow because elliptic-function periods can be algebraically related, like one of them being i times the other. For Jacobi elliptic functions, this is a result of the parameter being 1/2, which is algebraic, and its Weierstrass counterpart also has some algebraic counterpart.

Chudnovsky 1976: For Weierstrass elliptic functions, at least two of g2, g3, w1, w2, eta1, eta2 are algebraically independent.

Chudnovsky 1981: for g2, g3 algebraic, and u such that P(u) is algebraic, u is not a period, and linearly independent from w over the rational numbers. Then zeta(u) - eta(w)/w * u and eta(w)/w are algebraically independent.

With the result that pi/w and eta(w)/w are AI, and if the associated elliptic curve has "complex multiplication", then w and pi are AI.

Also, Gamma(1/3) and pi are AI, and Gamma(1/4) and pi are also AI. Also, at least two of pi, Gamma(1/5) and Gamma(2/5) are AI.


Then some discussion that I found very hard to follow, so I'll give some results.

pi, e^pi, Gamma(1/4) are AI
pi, e^(pi*sqrt(3)), Gamma(1/3) are AI


The paper ends with some unsolved problems, AI for:
pi, Gamma(1/3), Gamma(1/4)
at least three of pi, e^(pi*sqrt(5)), Gamma(1/5), Gamma(2/5)
e, pi, e^pi, Gamma(1/4)

Many of the functions I've mentions, and many other special functions, can be expressed as "hypergeometric functions".

 Hypergeometric function and  Confluent hypergeometric function and  

For these, I use the rising factorial or Pochhammer symbol
\( \displaystyle{ (x)_n = x (x+1) \cdots (x+n-1) = \sum_{k=0}^{n-1} (x+k) = \frac{\Gamma(x+n)}{\Gamma(x)} } \)

Plain ones are
\( \displaystyle{ {}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!} } \)

with differential equation
\( \displaystyle{ z(1-z) \frac{d^2 w}{dz^2} + (c - (a+b+1) z) \frac{dw}{dz} - a b w = 0 } \)

and integral form
\( \displaystyle{ \Beta(b,c-b) \, {}_2F_1(a,b;c;z) = \int_0^1 t^{b-1} (1-t)^{c-b-1} (1-z t)^{-a} dt } \)

and numerous interrelationships for different parameter values.

Confluent ones are
\( \displaystyle{ {}_1F_1(a;b;z) = \sum_{n=0}^\infty \frac{(a)_n}{(b)_n} \frac{z^n}{n!} } \)

with differential equation
\( \displaystyle{ z \frac{d^2 w}{dz^2} + (b - z) \frac{dw}{dz} - a w = 0 } \)

and integral form
\( \displaystyle{ \Beta(a,b-a) \, {}_1F_1(a;b;z) = \int_0^1 t^{a-1} (1-t)^{b-a-1} e^{z t} dt } \)

and also numerous interrelationships for different parameter values.

These functions can be generalized to arbitrary numbers of parameters:
\( \displaystyle{ {}_pF_q(a_1\dots a_p;b_1\dots b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!} } \)

with differential equation
\( \displaystyle{ \prod_{k=1}^p \left( z \frac{d}{dz} + a_k \right) w = \frac{d}{dz} \prod_{k=1}^q \left( z \frac{d}{dz} + b_k - 1 \right) w } \)

and also numerous interrelationships for different parameter values.

Some degenerate cases:

\( \displaystyle{ {}_2 F_1(a,b;b;z) = {}_1 F_0(a;;z) = (1 - z)^{-a} } \)

\( \displaystyle{ {}_1 F_1(a;a;z) = {}_0 F_0(;;z) = e^z } \)

More generally if a parameter is equal in the upper and lower parameter sets of a generalized hypergeometric function, it drops out.

A special value:

\( \displaystyle{ {}_2 F_1(a,b;c;1) = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)} } \)

Also,

\( \displaystyle{ \lim_{b\to\infty} {}_2 F_1(a,b;c;z/b) = {}_1 F_1(a;c;z} \)

\( \displaystyle{ \lim_{b\to\infty} {}_1 F_1(a;b;b z) = {}_1 F_0(a;;z) = (1-z)^{-a} } \)

For a generalized function, if an upper parameter is 0, then the function is 1. If an upper parameter is a negative integer, then the function is a polynomial in z.

Likewise, for a generalized function, if z -> z/a for some upper parameter a and a -> oo, that parameter then drops out. Likewise, if z -> a*z for some lower parameter a and a -> oo, that parameter then drops out.