log log log

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Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers

Volume VI Donal F. Connon

13 October 2007

Abstract

In this paper, the last of a series of seven, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function, including the following:

0

( 1) !

( , ) 1

n k n

k

B k S n k

= k

= −

∑

+

1 1 2

0

1 log(2) log(2 ) log log

1 2

n n

x n

x x dx n

− ⎛ ⎞⎜ ⎟ = −

+ ⎝ ⎠

∫

1 2

1 0

1 1 1

log log log ( ) ( ) 2 ( ) ( ) + ( ) ( )

1

q

a a a

dx q q q q q q

x ⁻ ^{⎛ ⎞}⎜ ⎟x ^⎡⎢ ^{⎛ ⎞}⎜ ⎟x ^⎤⎥ = Γ′′ ς + Γ′ ς′ Γ ς′′

+ ⎝ ⎠⎣ ⎝ ⎠⎦

∫

1

γ +2n ² ¹ ²₂

1 2

N k

k k

B kn

+

=

−

∑

^<^Hⁿ⁻^logⁿ^<^γ ⁺_2n¹ ² ²2 12

N k

k k

B

= kn

−

∑

( )

⁽¹⁾ ² ² ²

1 1 1

lim log log

2 ⁿ 2 2

n H γ n n γ

→∞

⎡ − − ⎤=

⎢ ⎥

⎣ ⎦

( )

⁽¹⁾ ² ⁽²⁾ ₍₁₎ ₍₂₎

1 1

lim 4 (3)

3

n n

k k

n n

n k k

H H

k k H H ς

→∞ = =

⎡ ⎤

⎢ + − ⎥=

⎢ ⎥

⎣

∑ ∑

⎦

( )

⁽¹⁾ ³ ⁽¹⁾ ⁽²⁾ ³ ²

(

²

)

³

1 1 1 1 1 1

lim log log (2) log (2)

6 ⁿ 2 ⁿ ⁿ 6 2 2 2 6

n H H H n γ n ς γ n ς γ γ

→∞

⎡ + − − − + ⎤= +

⎢ ⎥

⎣ ⎦

2

log ( ) log ( 1) ( )

k

k k

x x x k x

γ ^∞ k ς

=

Γ + + =

∑

−

Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

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CONTENTS OF VOLUMES I TO VI: Volume/page SECTION:

1. Introduction I/11 2. An integral involving cotx I/17

The Riemann-Lebesgue lemma I/21 3. Results obtained from the basic identities I/24 Some stuff on Stirling numbers of the first kind I/69 Euler, Landen and Spence polylogarithm identities I/86 An application of the binomial theorem I/117 Summary of harmonic number series identities I/154 4. Elementary proofs of the Flajolet and Sedgewick identities II(a)/5

Some identities derived from the Hasse/Sondow equations II(a)/13 A connection with the gamma, beta and psi functions II(a)/26 An application of the digamma function to the derivation

of Euler sums II(a)/31 Gauss hypergeometric summation II(a)/48 Stirling numbers revisited II(a)/51 Logarithmic series for the digamma and gamma functions II(a)/64 An alternative proof of Alexeiewsky’s theorem II(a)/74 A different view of the crime scene II(a)/91 An easy proof of Lerch’s identity II(a)/109 Stirling’s approximation for log ( )Γ u II(a)/114 The Gosper/Vardi functional equation II(a)/125

A logarithmic series for logA II(a)/137

Asymptotic formula for log (G u+1) II(a)/139

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The vanishing integral II(a)/147 Another trip to the land of G II(a)/165 Evaluation of

0

cot

x

un u du

π π

∫

II(a)/169 An observation by Glasser II(a)/188 An introduction to Stieltjes constants II(b)/5 A possible connection with the Fresnel integral II(b)/16 Evaluation of some Stieltjes constants II(b)/21 A connection with logarithmic integrals II(b)/67 A hitchhiker’s guide to the Riemann hypothesis II(b)/84 A multitude of identities and theorems II(b)/101 Various identities involving polylogarithms III/5 Sondow’s formula for γ III/42 Evaluation of various logarithmic integrals III/61 Some integrals involving polylogarithms III/66 A little bit of log( / 2)π III/88 Alternative derivations of the Glaisher-Kinkelin constants III/108 Some identities involving harmonic numbers IV/5

Some integrals involving log ( )Γ x IV/77 Another determination of log (1/ 2)G IV/88 An application of Kummer’s Fourier series for log (1Γ +x) IV/92

Some Fourier series connections IV/109 Further appearances of the Riemann functional equation IV/124 More identities involving harmonic numbers and polylogarithms IV/133 5. An application of the Bernoulli polynomials V/5

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6. Trigonometric integral identities involving:

- Riemann zeta function V/7 - Barnes double gamma function V/31 - Sine and cosine integrals V/45 - Yet another derivation of Gosper’s integral V/108 7. Some applications of the Riemann-Lebesgue lemma V/141 8. Some miscellaneous results V/145

APPENDICES (Volume VI):

A. Some properties of the Bernoulli numbers and the Bernoulli polynomials

B. A well-known integral

C. Euler’s reflection formula for the gamma function and related matters

D. A very elementary proof of

2

2 0

1 8 n (2n 1) π ^∞

=

∑

+

E. Some aspects of Euler’s constant γ and the gamma function F. Elementary aspects of Riemann’s functional equation for the zeta function

ACKNOWLEDGEMENTS REFERENCES

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APPENDIX A

SOME PROPERTIES OF THE BERNOULLI NUMBERS AND THE BERNOULLI POLYNOMIALS

As stated in (1.9) of Volume I, the Bernoulli polynomials B_n(x) are defined by the series

(A.1)

0

1 ( ) !

tx n

t n n

te t

e B x n

∞

=

− =

∑

, (t < 2π) The Bernoulli numbers B_n are given by the generating function (A.2)

!

1 ₀ n

B t e

t ⁿ

n

t

∑

^∞ n

=

− = , (t < 2π)

and it is readily seen from (A.1) that B_n(0)=B_n.

Using the Cauchy product [90, p.146] of two infinite series, equation (A.1) may be written as follows

0 0 0

( ) ! 1 ! !

n n n

tx n

n t n

n n n

t t t t

B x e B x

n e n n

∞ ∞ ∞

= = =

⎛ ⎞⎛ ⎞

⎛ ⎞

=⎜⎝ − ⎟⎠ = ⎜⎝ ⎟⎜⎠⎝ ⎟⎠

∑ ∑ ∑

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

=

∑ ∑

−

=

∞ −

= n

k

k n k n

n

k n k

x t B

0

0 !( )!

and equating coefficients of tⁿ we have (A.3)

0

( ) ⁿ ^{n k}

n k

k

B x n B x

k

−

=

= ⎛ ⎞⎜ ⎟

∑

⎝ ⎠

This identity could also be obtained in a less rigorous manner by formally

differentiating (A.1) with respect to x. It is therefore evident from (A.3) that B_n(x) is a polynomial of degree n and that the coefficient of xⁿ is 1 (i.e.,B_n(x) is a monomial because B₀ =1).

Using the identity

_t ^xt

xt t

t x

e te te e

te =

− −

−

+

1 1

) 1 (

and (A.1) we have

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! ) !

! ( ) 1 (

0 0

0 n

x t n

x t n B

x t B

n

n n n

n

∑ ∑

∑

^∞

=

∞

=

∞

=

− +

Equating coefficients of tⁿ we obtain

(A.4) B_n(1+x)−B_n(x)=nxⁿ⁻¹ for n≥1. Letting x=0 in (A.4) we get

(A.5) B_n(1)=B_n(0)=B_n for n≥2. Now, substituting x=1 in (A.3) we have for n≥2 (A.6)

0 n

n k

k

B n B

= k

= ⎛ ⎞⎜ ⎟

∑

⎝ ⎠ Using L’Hôpital’s rule we have 1

lim lim 1

1

x x

x o x o

x

e e

→ = → =

−

and we therefore conclude that B₀ =1. Simple algebra shows that

⎟⎟=

⎠

⎜⎜ ⎞

⎝

⎛ −

−

= +

− 1

1 1 2

1 ^x

x

x e

e x e

x ⎟⎟=

⎠

⎜⎜ ⎞

⎝

⎛

− + +

− 1

1 2

2 ^x

x

e e x x

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

− + +

− ₋

−

2 2

2

2 ^x ^x

x x

e e

e e x x

and therefore using (A.2) we have

(A.7) coth( /2) 2

!

2 ₀ x x

n B x

x ⁿ

n

n =

+

∑

^∞

=

The right hand side of (A.7) is an even function of x and hence B₂_n₊₁=0 for all n≥1and B₁=

2

−1

It would be possible to determine B₁ using the Maclaurin series expansion of (A.2):

however, using the expansion to determine other coefficients is difficult because of the complexity introduced by computing the higher derivatives. We can however use the recursion formula (A.6) to obtain succeeding values of B_n as follows:

(A.8) B₀ =1

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2 1

1 =− B

6 1

2 = B

30 1

4 =− B

42 1

6 = B

From the method of recursive calculation shown by (A.6), it can immediately be seen that all of the B_n are rational numbers. Since (2 )ς n is always positive, we can see from (1.7) that the numbers B_2n alternate in sign, i.e.

(A.9) (−1)ⁿ⁺¹B₂_n> 0 From (1.1) we have

(A.10) ^a _n

k n

n

n k ₋

∞

=

−

− − = −

= −

∑

1

1 1 2

) ( )

1 ( 2

1 ) 1

( ς

ς

Since )ς_a(n is an alternating series, we have the inequality _n

2

1− 1 < ς_a(n) < 1

and hence we have the bounds for ς( )n _n

n

−

− 21

1 2

1 < ( )ς n < ₋_n

−2¹ 1

1

Therefore, it is apparent that ς( )n →1 as n→ ∞.

In [118a] Sasvári gave an elementary proof of Binet’s formula for the gamma function

( 1) 2 . ^{( )}

x

x x

x x e

e

π ϑ

Γ + = ⎜ ⎟⎛ ⎞⎝ ⎠

where

0

1 1 1

( ) 1 2

xt t

x e dt

e t t

ϑ =^∞

∫

^⎛⎜⎝ − − + ^⎞⎟⎠ ⁻ Since lim ( ) 0

x ϑ x

→∞ = , we immediately obtain Stirling’s asymptotic formula

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(A.11) !n = Γ +(n 1)~ 2 n n

e πn

⎛ ⎞⎜ ⎟

⎝ ⎠ and we have

2 12 1

( 1) 2 ~ 4

n n

n

B e n

π e π

+

+ ⎛ ⎞

− ⎜⎝ ⎟⎠

Hence the absolute value of B_2n grows rapidly in size. In fact, B₄₈~ 1.20866 x 10 ²³ and this is of the same order of magnitude as Avogadro’s number N ~ 6.022 x 10²³. Indeed B₁₀₂ is approximately equal to the number of baryons (10⁸⁰) in the observable universe [106, p.728].

Since the left hand side of (A.1), with x=1/2, is an even function of t, we deduce (A.12) B₂_n₊₁(1/ 2) 0= for n ≥ 0.

Differentiating (A.3) it is seen that (A.13) B_n′(x)=nB_n₋₁(x) We also have

(A.14) B_n(1−x)=(−1)ⁿB_n(x) B_n(1+ −x) B x_n( )=nxⁿ⁻¹ Since B₂_n(1−x)=B₂_n( )x , we have (A.14a) B₂_n(1)=B₂_n(0)=B₂_n.

Similarly, B₂_n₊₁(1−x)= −B₂_n₊₁( )x implies that (A.14b) B₂_n₊₁(1)= −B₂_n₊₁(0)= −B₂_n₊₁ =0.

With knowledge of B_n we can compute the Bernoulli polynomials B_n(x) using (A.3).

The first few are:

(A.15) B x₀( ) 1= ₁ 1

( ) 2

B x = −x

₂ ² 1

( ) 6

B x =x − +x

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₃ ³ 3 ² 1 1

( ) ( 1)(2 1)

2 2 4

B x =x − x + x= x x− x−

₄ ⁴ ³ ² 1

( ) 2

B x =x − x +x −30

Since B₂_n₊₁ =0, equation (A.7) can be written as

(A.16) ₂ ²

0

coth( / 2)

2 (2 )!

n n n

x x

x B

n

∞

=

∑

, (x < 2π)

and, since cotx=icothix, ( i= −1), replacing x by 2ix we have the identity which was employed in Volume V

(A.17) ⁿ ⁿ

n

n x

n x B

x ² ²

2

0 (2 )!

) 2 1 ( cot

∑

^∞

=

−

= , ( x < π)

The following identity is easily derived

(A.18) x

x x

x x x

x x

x 2cot2

cos sin

sin cos

cos sin sin

tan cos cot

2

2 − =

=

−

=

−

and this enables us to write

(A.19) ,

)!

2 (

) 1 2 ( ) 2 1 (

tan ² ² ² ² ¹

1

1 −

∞

=

+ −

−

=

∑

ⁿ ⁿ ⁿ ⁿ

n

n x

n

x B (x < π/2)

Since (A.20)

x x

x sin2

tan 2

cot + =

we can also easily show that

(A.21) ⁿ ⁿ

n

n x

n B x

x ² ₂ ₂

0 1

)!

2 (

) 2 2 ) ( 1 sin (

− −

=

∑

^∞

=

+

Note that identity (A.20) is incorrectly recorded in Knopp’s book [90, p.208].

We also have [90, p.239]

(A.22) ⁿ ⁿ

n

n x

n E x

2 2 0( 1) (2 )!

cos

1

∑

^∞

=

−

=

where E_2n are the Euler numbers defined in (A.26) below.

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Since cot( / 2)x → ∞ as x→2π, one would suspect that the radius of convergence of (A17) would be 2π. This is shown to be the case in [90, p.237] and [66, p.586]. The radius of convergence is equal to 2π because the nearest singularities of z/(e^z−1)are

2πi

± .

We also have an explicit formula for B_n given by F. Lee Cook in [47]: see also Gould’s paper [73a], Rademacher’s book [110a, p.9] and the short paper published by Rzadkowski [116a] in 2004. In what follows, I have applied Lee Cook’s approach to the Bernoulli polynomial, rather than to B_n .

Using (A.1) and the Maclaurin expansion we have

0

( ) 1

n tx

n n t

t

d te

B x dt e ₌

⎡ ⎛ ⎞⎤

= ⎢⎣ ⎜⎝ − ⎟⎠⎥⎦

Since log 1 (1t= ⎡⎣ − −e^t)⎤⎦ and

1

log(1 ) for

m

u u u

m

∞

=

− = −

∑

< 1, we have

1

(1 ^{t m})

m

t e

m

∞

=

= −

∑

−

Therefore, dividing by (e^t−1) we get

1

tx t

te

e =

−

1

1 0

(1 ) (1 )

1

tx t m tx t k

m k

e e e e

m k

∞ − ∞

= =

− −

= +

∑ ∑

^for^t ^{< log2}

Hence we have

0

( ) 1

n 1

k

B x k

∞

=

∑

+

( )

0

1

n

t tx n

t

d e e

dt ₌

⎡ ⎤

⎢ − ⎥

⎣ ⎦

Using the Leibniz rule for the derivative of a product we note that the nth derivative vanishes at t=0 for k≥ +n 1. Therefore, using the binomial theorem we have

⁽ ⁾

0 0 0

( ) 1 ( 1) ( )

1

k n

j j x t

n n

k j t

k d

B x e

k j dt

∞ +

= = =

⎡ ⎤

=

∑

+

∑

− ⎛ ⎞⎜ ⎟ ⎢⎝ ⎠ ⎣ ⎥⎦

Hence we have (A.23)

0 0

( ) 1 ( 1) ( )

1

n k

j n

n

k j

B x k x j

j

= k =

=

∑

+

∑

− ⎛ ⎞⎜ ⎟⎝ ⎠ +

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A different proof, using the Hurwitz-Lerch zeta function, was recently given by Guillera and Sondow [75aa]. They also noted that

0

( 1) ( ) 0

k

j n

j

k x j

= j

− ⎛ ⎞⎜ ⎟ + =

∑

⎝ ⎠ ^for^k^>ⁿ⁼^{0,1, 2,...}

and we therefore have (A.23i)

0 0

( ) 1 ( 1) ( )

1

k

j n

n

k j

B x k x j

k j

∞

= =

=

∑

+

∑

− ⎛ ⎞⎜ ⎟⎝ ⎠ +

When x=0 we obtain (A.23aa)

0 0 0 0

1 1

(0) ( 1) ( 1)

1 1

n k k

j n j n

n n

k j k j

k k

B B j j

j j

k k

∞

= = = =

⎛ ⎞ ⎛ ⎞

= =

∑

+

∑

− ⎜ ⎟⎝ ⎠ =

∑

+

∑

− ⎜ ⎟⎝ ⎠

Note the structural similarity of (A.23aa) with the Stirling numbers of the second kind which we briefly referred to in (3.97) et seq. of Volume I. In particular (3.100) shows that

j 0

( , ) 1 ( 1)

!

k

k j k n

S n k j

j k

−

=

= − ⎛ ⎞⎜ ⎟

∑

⎝ ⎠ We therefore conclude that

(A.23a)

0

( 1) !

( , ) 1

n k n

k

B k S n k

= k

= −

∑

+

Surprisingly, this simple relationship does not appear in the book “Concrete Mathematics” [75]: instead, the authors report a much more complex identity involving both kinds of the Stirling numbers [75, p.289]. I subsequently discovered that the formula (A.23a) is reported in the Wolfram Mathworld website dealing with the Bernoulli numbers. Furthermore, this identity was also proved by Kaneko in 2000 in his paper “The Akiyama-Tanigawa algorithm for Bernoulli numbers” [82a], albeit his method was somewhat less direct. Interestingly, in that paper Kaneko reports that the Bernoulli numbers were independently discovered by Takakazu Seki (1642-1708) one year prior to Jakob Bernoulli. A proof was also given by Akiyama and Tanigawa in [6ai]. That paper also mentions that

(A.23b)

1 1

( , ) ( , )

k k

k j k j

j j

a s k j b b S k j a

= =

=

∑

⇔ =

∑

and hence we obtain (A.23c)

1

( 1) ! ( , )

1

k k

r r

s k r B k

= k

= −

∑

+

In fact, the Bernoulli numbers were originally introduced by Johann Faulhaber (1580- 1635) in his book Academia Algebrae published in 1631. To his credit, Jakob

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Bernoulli (1654-1705) did in fact give priority to Faulhaber in his treatise on probability theory, Ars Conjectandi, which was published posthumously in 1713.

Euler invented the Euler numbers [62] to study the sums (A.24) ⁿ ^k

j

k n j

T

∑

⁻

=

−

= ¹

0

) 1 ( ) (

The Euler Polynomials E_n(x) are defined in [126, p.63] by means of the generating function

(A.25) t n

x t e E

e ⁿ

n t n

xt

( , ) !

1 ( 2

∑

^∞0

=

+ = < π )

and the Euler numbers are defined by (A.26) E_n =2ⁿE_n(1/2) The Dirichlet Beta function is defined by (A.27)

∑

^∞

= +

= −

0(2 1) ) 1 ) (

(

n

s n

s n β

and for s=2, (2)β is known as Catalan’s constant G ₂

0

( 1) 0.915965...

(2 1)

n

G n

∞

=

= − =

∑

+

It is still not known if G is irrational (or indeed transcendental).

We have the relationship for the Dirichlet beta function (A.28)

)!

2 ( 2

) 2 / ( ) 1 ) ( 1 2

( ²

1 2

k

k E ^k

k

k +

= −

+ π

β

The Dirichlet beta function was employed in the recent paper by Dalai [51].

Other miscellaneous identities are set out below for ease of reference.

(A.29) ² ²

0

sec (2 )!

n n

n

x E x

n

∞

=

∑

x < π/2 and using (A.28) we have

(A.30) ¹ ²

0

sec( ) 4ⁿ (2 1) ⁿ

n

x n x

π π ^∞ ⁺ β

=

∑

+

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APPENDIX B

A WELL-KNOWN INTEGRAL

We recall the famous dictum [133, p.123] of the renowned Irish mathematician and physicist, Lord Kelvin (William Thomson (1824-1907)), that

“A mathematician is a person to whom

0 2

2 ₌ π

∫

∞

− dx

e ^x is as obvious as 1+1 = 2”.

A very elegant proof of the above identity, presented by H.F. Sandham [117] in 1946, is set out below.

Let

⁼^∞

∫

⁻

0

2

dt e

I ^t

and, using the substitution t=xy, we obtain ^{2 2}

0

I e x y y dx

∞ −

=

∫

Therefore we have

² ²⁽¹ ²⁾

0

2e ^y I e ^y ^x 2y dx

∞

− =

∫

− +

Integrating again from 0 to ∞with respect to y ² ²⁽¹ ²⁾

0 0 0

2I e ^y dy dy e ^y ^x 2y dx

∞ ∞ ∞

− = − +

∫ ∫ ∫

and, using Fubini’s theorem to interchange the order of integration, we have ²⁽¹ ²⁾

0 0

2

y x

dx e y dy

∞ ∞

− +

=

∫ ∫

₁ ^tan 0 ₂

1 0

2

=π + =

=^∞

∫

^dx_x ⁻ ^x^∞ Therefore we get 2I² =π/2, and hence we have the important identity

0 2

2 ₌ π

∫

∞

− dx e ^x

I subsequently discovered that a similar proof was given by J.-A. Serret in his 1900 tome “Cours de Calcul Différentiel et Intégral”, Vol. 2, Calcul Intégral (p.132), a copy of which may be viewed on the internet at The Cornell Library Historical

Mathematics Monographs website.

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APPENDIX C

EULER’S REFLECTION FORMULA AND RELATED MATTERS

Theorem: Euler’s reflection formula for the gamma function states that

(C.1) ( ) (1 ) x x sin

x π Γ Γ − = π Proof:

The following is based on a proof provided by Dedekind [52] in 1853 (as outlined in an exercise in the recent book “Special Functions” [8a, p.49] by Andrews et al.).

Various intermediate steps of this proof give rise to interesting identities in their own right.

Let us put

(C.2) dt t x t

∫

x

∞ −

= +

0 1

) 1 φ(

where we require 0 < x < 1 for convergence (throughout this proof, I have found it useful to keep in mind the fact that we will eventually show that ( )

x sin

x φ π

= π ).

Using the substitution t=sy it is easily seen that (C.3) dt

st x t

s

x

∫

^∞ ⁻

−

= +

0 1

) 1

φ(

Now let s=1/p in (C.3) to obtain

(C.4) dt t p x t p

x x⁻ ⁼^∞

∫

₊⁻

0 1φ( ) 1

Or equivalently

(C.5) dt t s x t s

x x⁻ ⁼^∞

∫

₊⁻

0 1 1φ( )

Dividing (C.5) by (s+1) and integrating we obtain

(C.6) dt ds

t s

t ds s

s x s

x x

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

= +

+

∫ ∫

∫

^∞ ^∞ ⁻

∞ −

0 1

0 0

1

1 1 ) 1

φ( Using (C.2) we have

(15)

(C.7) dt ds t s

t x s

x

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

=

∫

^∞ +

∫

^∞ ⁻

0 1

0 2

1 ) 1

φ (

Changing the order of integration we obtain

(C.8) ds t s dt s

t

x ⁼

∫

^∞ ^x⁻

∫

^∞ ₊ ₊

0 0

1 2

) )(

1 ( ) 1

φ (

It is readily seen that (C.9)

0 0

1 1 1

( 1)( )ds (1 )( ) (1 )( 1) ds

s s t t s t t s

∞ ∞ ⎡ ⎤

= ⎢ − ⎥

+ + ⎣ − + − + ⎦

∫ ∫

(C.10)

0

1 1

log log

1 1 1

s t t

t s t

+ ∞

⎛ ⎞

= − ⎜⎝ + ⎟⎠ = −

From (C.8) we therefore have (C.11)

1 1

2

0 0

( ) log

1 1

x x

t t d t

x dt dt

t dx t

φ =^∞ ⁻ = ^∞ ⁻

− −

∫ ∫

We have the elementary integral

∫

t dy^x =t^x/ logt and hence integrating (C.11) with respect to x we have

(C.12) ² ¹

1 1 0

( ) log

1

y y x

y y

t t

x dx dx dt

φ ^∞ ⁻t

− −

= −

∫ ∫ ∫

(C.13) ¹

0 1

y y

t t

t dt

∞ − − −

=

∫

−

Now, subtracting (C.3) from (C.5) we obtain (C.14)

1 1

0

( ) ( 1)

( )

1 ( 1)( )

x x x

s s t t

x dt

s φ ^∞ st s t

− − − − −

− =

∫

+ +

We now integrate (C.14) with respect to s over (0,∞) and use (C.13) to obtain (C.15) ¹

0

( )

( ) 1

x x

s s

x ds

φ ^∞ ⁻s⁻ ⁻ =

∫

−

∫

⁼

−

dt t x

x

1 x 2( ) )

( φ

φ dt

t s st

t ds t

∫

x

∫

^∞ ⁻

∞

+ +

−

0 1

0 ( 1)( )

) 1 (

(16)

(C.16) dt t s dt st

t

t ^x

∫

^∞

∞ −

+

− +

=

0 0

1

) )(

1 ( ) 1

1 (

Using partial fractions it is easily shown that

(C.17) ds

st t

t t

s ds t

t s

st (1 )( ) (1 )( 1)

1 )

)(

1 (

1

2 0

2

0 − − +

+

= − +

+

∫

^∞

∞

(C.18) ₂ ₂

0

1 2

log log

1 1 1

s t t

t st t

+ ∞

⎛ ⎞

= − ⎜⎝ + ⎟⎠ = −

Substituting (C.18) in (C.16) we have (C.19) ²

1

( ) ( )

x

x t dt

φ φ

−

∫

= ¹

0

2 log 1 tx t

t dt

∞ −

∫

+

(C.20) =2φ′(x), using the definition of φ(x) in (C.1).

Substituting t=1/u in (C.2) it is easily seen that (C.21) dt

t x t

∫

x

∞ −

= +

0 1

) 1

φ( (1 )

01

x t dt

t ^x

− + =

=

∫

^∞ ⁻ ^φ

In addition, using the substitution t=1−u we have

(C.22)

12

2 2

1 1

( ) ( )

x

x x

t dt t dt

φ φ

− −

∫

=

∫

²

12

( )

x

φ t dt

+

∫

²

12

2 ( )

x

φ t dt

=

∫

Therefore, using (20) we obtain (C.23) ²

12

( ) ( ) ( )

x

x t dt x

φ

∫

φ =φ′

Differentiating (C.23) gives us the differential equation (C.24) φφ′′−

( )

φ′ ² =φ⁴

With 1/φ = uwe obtain

(C.24a) uu′′=( )u′ ²−1

and the solution of (C.24) with the initial conditions φ(1/2)=πand 0φ′(1/2)= is

(17)

(C.25) ( ) x sin

x φ π

= π

However, without actually knowing the solution, it is not immediately obvious to me how one would integrate (C.24) from first principles.

Now making the substitution t =s/(s+1)in the beta integral ⁼

∫

¹ ⁻ ⁻ ⁻

0

1 1(1 ) )

,

(x y t t dt

B ^x ^y

we obtain

(C.26)

1

0

( ) ( ) ( , )

(1 ) ( )

x x y

s x y

B x y ds

s x y

∞ −

+

Γ Γ

= =

+ Γ +

∫

Letting y=1−x in (C.26) we have (C.27)

) 1 (

) 1 ( ) ( ) 1

(

0 1

Γ

− Γ

= Γ

=

∫

^∞ ^s+⁻_s^ds ^x ^x x

φ x

We have therefore proved the theorem (C.28) ( ) (1 )

x x sin

x π

Γ Γ − = π SinceΓ + = Γ(x 1) x ( )x it is easily seen that this identity is in fact valid [128, p.149] for

all x∈C except for x=0 and all of the integers (positive and negative).

⁪

A number of interesting results automatically arise from various steps in the proof: for example, by differentiating (C.2) we obtain

(C.29)

2 1

2

0

cos log

sin 1

x tx t

x t dt

π π

π

∞ −

= −

∫

+

By differentiating (C.5) with respect to s we obtain (C.30) ² ¹ ₂

0

( 1)

sin ( )

x x

x s t

x s t dt π

π

− ∞ −

− = −

∫

+

Alternatively, by differentiating (C.5) with respect to x we obtain (C.31)

2 1

1

2

0

log cos log

sin sin

x

x s x t t

s dt

x x s t

π π π

π π

∞ −

− ⎛ ⎞

− =

⎜ ⎟ +

⎝ ⎠

∫

(18)

From (C.11) we have

(C.32)

2 1

2

0

log

sin 1

tx t x t dt π

π

∞ −

=

∫

−

and differentiating (C.32) with respect to x we obtain (C.33)

3 1 2

3

0

2 cos log

sin 1

x tx t

x t dt

π π

π

∞ −

= −

∫

−

⁪ We have

1 1 1 1

0 0 1

( ) 1 1 1

t t t

dt dt dt

t t t

α α α

φ α =^∞ ⁻ = ⁻ +^∞ ⁻

+ + +

∫ ∫ ∫

and using the substitution t→1/t in the last integral we obtain

1 1

0 0 0

1 1 1

0

1 1 1

, where =1 1

t t t

dt dt dt

t t t

t t

t dt

α α α

α β

β α

∞ − − −

− −

= +

+ + +

= + −

+

∫ ∫ ∫

∫

If 0 < α < 1 then we also have 0 < β< 1. Let us now consider the integral

1 1

0

( ) 1

t t

t dt

α α

φ α = ⁻ ⁺ ⁻

∫

+ Then, upon integration we have

1 1

0

( ) 1

x x

a a

t t

d d dt

t

α α

φ α α = α ⁻ ⁺ ⁻

∫ ∫ ∫

+

and, interchanging the order of integration, we obtain

1 1 1 1

0(1 ) log 0 (1 ) log

x x a a

t t t t

dt dt

t t t t

− − − − − −

= −

+ +

∫ ∫

This immediately reminded me of equation (4.4.112g) which was originally derived by Kummer and is valid for α > 0, β> 0.

(19)

1 1 1

0

1

2 2

log 1

(1 ) log

2 2

t t

t tdt

α β α β

β α

− −

⎛ + ⎞ ⎛ ⎞ Γ⎜ ⎟ ⎜ ⎟Γ

− = ⎝ ⎠ ⎝ ⎠

+ Γ⎛⎜⎝ + ⎞ ⎛ ⎞⎟ ⎜ ⎟⎠ ⎝ ⎠Γ

∫

As a result we have with β = −1 α

(C.33a)

1 1

0

1 1

2 2

log 2

(1 ) log

2 2

x x

t t

dt x x

t t

− −

+ −

⎛ ⎞ ⎛ ⎞

Γ⎜ ⎟ ⎜Γ ⎟

− = ⎝ ⎠ ⎝ ⎠

+ Γ⎛⎜ − ⎞ ⎛ ⎞⎟ ⎜ ⎟Γ

⎝ ⎠ ⎝ ⎠

∫

We therefore have ( )

x

a

φ α αd =

∫

^log^Γ^⎛^⎜_⎝¹⁺₂^x^⎞^⎟_⎠⁺^log^Γ^⎛^⎜_⎝¹⁻₂^x^⎞^⎟_⎠⁻^log^Γ^⎛^⎜_⎝²⁻₂^x^⎞^⎟_⎠⁻^log^Γ^{⎛ ⎞}^{⎜ ⎟}_{⎝ ⎠}₂^x

1 1 2

log log log log

2 2 2 2

a a a a

+ − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

− Γ⎜ ⎟− Γ⎜ ⎟+ Γ⎜ ⎟+ Γ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

In (C.25) ( )φ α was defined as ( )

sin φ α π

= πα and we have the elementary integral

sec (2 / 2) 1

log tan( / 2) sin 2sin( / 2) cos( / 2) 2 tan( / 2)

dx dx x

dx x

x x x x

π π

π ⁼ π π ⁼ π ⁼π

∫ ∫ ∫

Hence we have

( )

x

a

φ α αd =

∫

^{log tan(} / 2) log tan( / 2)

sin

x

a

dα x a

π π π

πα ⁼ ⁻

∫

and therefore we get

(C.34) log tan(πx/ 2) log tan(− πa/ 2)=

1 1 2

log log log log

2 2 2 2

x x x x

+ − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

Γ⎜⎝ ⎟⎠+ Γ⎜⎝ ⎟⎠− Γ⎜⎝ ⎟⎠− Γ⎜ ⎟⎝ ⎠

1 1 2

log log log log

2 2 2 2

a a a a

+ − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

− Γ⎜ ⎟− Γ⎜ ⎟+ Γ⎜ ⎟+ Γ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(20)

where we require 0 < a < x < 1 , or more generally n < a < x < n+1 with n being a strictly positive integer to ensure that sinπα ≠ ∀ ∈0 α [ , ]a x .

⁪ We can write

1 1 2

log log log log

2 2 2 2

x x x x

+ − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

Γ⎜⎝ ⎟⎠+ Γ⎜⎝ ⎟⎠− Γ⎜⎝ ⎟⎠− Γ⎜ ⎟⎝ ⎠

1 1

log log log 1 log

2 2 2 2 2 2

x x x x

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= Γ⎜ + ⎟+ Γ⎜ − ⎟− Γ −⎜ ⎟− Γ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

log (1= Γ −y) log ( ) log (1+ Γ y − Γ − −z) log ( )Γ z

where 1

and .

2 2 2

x x

y= − z=

⁼^{log (1}

[

^{Γ −}^y^{) log ( )}^Γ ^y

]

⁻^{log (1}

[

^{Γ −}^z^{) log ( )}^Γ ^z

]

log log

sin y sin z

π π

= −

= −log sinπy+log sinπz

Since 1

sin sin cos

2 2 2

x x

y π

π = π^⎡⎢⎣ − ^⎤⎥⎦= we obtain

(C.35) 1 1 2

log log log log log tan

2 2 2 2 2

x x x x πx

+ − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

Γ⎜⎝ ⎟⎠+ Γ⎜⎝ ⎟⎠− Γ⎜⎝ ⎟⎠− Γ⎜ ⎟⎝ ⎠= ⎜⎝ ⎟⎠

and therefore (C.36)

1 1

2 2

2

2 2

x x

+ −

⎛ ⎞ ⎛ ⎞

Γ⎜⎝ ⎟ ⎜⎠ ⎝Γ ⎟⎠

⎛ − ⎞ ⎛ ⎞ Γ⎜ ⎟ ⎜ ⎟Γ

⎝ ⎠ ⎝ ⎠

tan 2 πx

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

The above result may be obtained more directly from (C.34) by letting a=1/ 2. We then see from (C.33a) that

(C.36a)

1 1

0

log tan

(1 ) log 2

x x

t t x

t tdt

π

− − − = ⎛⎜ ⎞⎟

+ ⎝ ⎠

∫

Letting x=3 / 2 we have

(21)

1 1

1 4 4

1 1

4 1 4

⎛ ⎞ ⎛ ⎞

Γ +⎜ ⎟ ⎜Γ − ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ Γ⎜ ⎟ ⎜⎝ ⎠ ⎝Γ − ⎟⎠

tan 3 4

⎛ π ⎞

= ⎜⎝ ⎟⎠

Using Euler’s functional equation we have (1Γ −x)= − Γ −x ( x) and hence 1 1 1

1 4 4 4

⎛ ⎞ ⎛ ⎞

Γ −⎜ ⎟= − Γ −⎜ ⎟

⎝ ⎠ ⎝ ⎠

1 1

4 1 4

⎛ ⎞

Γ +⎜⎝ ⎟⎠ Γ⎜ ⎟⎛ ⎞⎝ ⎠

1 3 1

4tan 4 4

⎛ π ⎞

= − ⎜ ⎟=

⎝ ⎠

This is a particular case of the more general identity

1

1 1 1

1 (4 3)

4 4 4

n n

k

n k

=

⎛ ⎞ ⎛ ⎞

Γ⎜ + ⎟= Γ +⎜ ⎟ −

⎝ ⎠ ⎝ ⎠

∏

From (C.36), and using Euler’s reflection formula in the denominator, we have (as recorded in [74, p.887])

(C.37) 1 1

2 2

x x

+ −

⎛ ⎞ ⎛ ⎞

Γ⎜⎝ ⎟ ⎜⎠ ⎝Γ ⎟⎠ sec 2 πx

π ^⎛ ^⎞

= ⎜⎝ ⎟⎠

and letting x=1/ 2 we have (C.37a) 3 1

4 4

⎛ ⎞ ⎛ ⎞

Γ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠Γ sec 2 4

π ^{⎛ ⎞}π π

= ⎜ ⎟⎝ ⎠=

(this is in numerical agreement with the approximations contained in [1, p.255] for

3 1

and

4 4

⎛ ⎞ ⎛ ⎞

Γ⎜ ⎟ Γ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ). According to Havil [78, p.55], no closed form expression

for 1

4

Γ⎜ ⎟⎛ ⎞⎝ ⎠is yet known.

We have Legendre’s relation [66, p.493] for x > 0 (C.37b) 1 ₁

2 2 2^x ( )

x x

π x

−

⎛ ⎞ ⎛ + ⎞

Γ⎜ ⎟ ⎜⎝ ⎠ ⎝Γ ⎟⎠= Γ

and for x=1/ 2 we get the same result as (C.37a) . With x→2x we get

(22)

( )

¹ ₂ ₁ ^{(2 )}

2 2 ^x

x x π x

−

⎛ ⎞

Γ Γ⎜⎝ + ⎟⎠= Γ Rudin [115, p.194] gives a short proof of Legendre’s relation using the Bohr- Mollerup theorem [25, p.187]. Harald Bohr (1887-1951), who was the younger brother of the renowned Danish physicist Niels Bohr, was one of the most prominent Danish mathematicians in the first half of the 20th century (as well as being an international footballer who played in the 1908 Olympics for Denmark, beating France by the remarkable score of 17 to 1!). It is also possible to prove the Euler reflection formula by applying the Bohr-Mollerup theorem to the function ( )

(1 )sin

f x x x

π

= π

Γ −

⁭ From (C.13) we have

²

1

( )x dx

α

φ

−

∫

¹

0 1

t t

t dt

α α

∞ − − −

=

∫

−

Since 1₂ 1

sin dx cot x

x π

π ^{= −}π

∫

^{we have}

₂²

[ ]

1

cot (1 ) cot sin dx

x

α

π π π α πα

− π

= − −

∫

¹

0 1

t t

t dt

α α

∞ − − −

=

∫

−

We have

1 1 1 1

0 1 0 1 1 1

t t t t t t

dt dt dt

t t t

α α α α α α

∞ − − − − − − ∞ − − −

= +

− − −

∫ ∫ ∫

and using the substitution t→1/t in the second integral we obtain

1 1

1 1 1

0 0 0

1 1

0

1 1 1

2

1

t t t t t t

dt dt dt

t t t

t t

t dt

α α α α α α

α α

∞ − − − − − −

− −

− − −

= −

− − −

= −

−

∫ ∫ ∫

∫

and therefore we have

(C.38) ^π

[

^{cot (1}^π ⁻^α^{) cot}⁻ ^πα

]

¹ ¹

0

2 1

t t

t dt

α− − −α

=

∫

−