Part 10: The digamma function

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Series A: Mathematical Sciences, Vol. 17 (2009), 45–66 Universidad T´ecnica Federico Santa Mar´ıa

Valpara´ıso, Chile ISSN 0716-8446

c Universidad T´ecnica Federico Santa Mar´ıa 2009

The integrals in Gradshteyn and Ryzhik.

Part 10: The digamma function

Luis A. Medina and Victor H. Moll

Abstract. The table of Gradshteyn and Rhyzik contains some integrals that can be expressed in terms of the digamma function ψ(x) = dx^d log Γ(x). In this note we present some of these evaluations.

1. Introduction

The table of integrals [2] contains a large variety of definite integrals that involve the digamma function

(1.1) ψ(x) = d

dxlog Γ(x) = Γ^′(x) Γ(x). Here Γ(x) is the gamma function defined by

(1.2) Γ(x) =

Z ∞ 0

t^x−1e^−tdt.

Many of the analytic properties can be derived from those of Γ(x). The next theorem represents a collection of the important properties of Γ(x) that are used in the current paper. The reader will find in [1] detailed proofs.

Theorem 1.1. The gamma function satisfies:

a) the functional equation

(1.3) Γ(x + 1) = xΓ(x).

b) For n ∈ N, the interpolation formula Γ(n) = (n − 1)!.

2000 Mathematics Subject Classification. Primary 33.

Key words and phrases. Integrals, Digamma function.

The second author wishes to acknowledge the partial support of NSF-DMS 0409968. The first author was partially supported as a graduate student by the same grant.

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c) The Euler constant γ, defined by

(1.4) γ = lim

n→∞

n

X

k=1

1 k− ln n,

is also given by γ = −Γ^′(1). This appears as the special case a = 1 of formula 4.331.1:

(1.5)

Z ∞ 0

e^−axln x dx = −γ + ln a

a .

This was established in [3]. The change of variables t = ax shows that the case a = 1 is equivalent to the general case. This is an instance of a fake parameter.

d) The infinite product representation

(1.6) Γ(x) = e^−γx

x

∞

Y

k=1

1 + x

k

−1

e^x/k

is valid for x ∈ C away from the poles at x = 0, −1, −2, . . . e) For n ∈ N we have

(1.7) Γ n +¹₂ = (2n)!

2²ⁿn!

√π

and

(1.8) Γ ¹₂− n = (−1)ⁿ2²ⁿn!

(2n)!

√π.

f) For x ∈ C, x 6∈ Z we have the reflection rule

(1.9) Γ(x)Γ(1 − x) = π

sin πx.

Several properties of the digamma function ψ(x) follow directly from the gamma function.

Theorem 1.2. The digamma function ψ(x) satisfies a) the functional equation

(1.10) ψ(x + 1) = ψ(x) + 1

x. b) For n ∈ N, we have

(1.11) ψ(n) = −γ +

n−1

X

k=1

1 k. In particular, ψ(1) = −γ.

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c) For x ∈ C away from x = 0, −1, −2, · · · we have ψ(x) = −γ − 1

x+ x

∞

X

k=1

1 k(x + k) (1.12)

= −γ −

∞

X

k=0

1

x + k− 1 k + 1

d) The derivative of ψ is given by

(1.13) ψ^′(x) =

∞

X

k=0

1 (x + k)². In particular, ψ^′(1) = π²/6.

e) For n ∈ N we have

(1.14) ψ(¹₂± n) = −γ − 2 ln 2 + 2

n

X

k=1

1 2k − 1. In particular,

(1.15) ψ(¹₂) = −γ − 2 ln 2.

f) For x ∈ C, x 6∈ Z we have the reflection rule

(1.16) ψ(1 − x) = ψ(x) + π cot πx.

2. A first integral representation

In this section we establish the integral evaluation 3.429. Several direct conse- quences of this formulas are also described.

Proposition 2.1. Assume a > 0. Then (2.1)

Z ∞ 0

e^−x− (1 + x)^−a dx

x = ψ(a).

Proof. We begin with the double integral (2.2)

Z ∞ 0

Z s 1

e^−tzdt dz = Z ∞

0

e^−z− e^−sz

z dz.

On the other hand, (2.3)

Z s 1

Z ∞ 0

e^−tzdz dt = Z s

1

dt t = ln s.

We conclude that (2.4)

Z ∞ 0

e^−z− e^−sz

z dz = ln s.

This evaluation is equivalent to:

(2.5)

Z ∞ 0

e^−ax− e^−bx

x dx = lnb a,

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that appears as formula 3.434.2 in [2]. The reader will find a proof in [3].

We now establish the result: start with Γ^′(a) =

Z ∞ 0

e^−ss^a−1ln s ds

= Z ∞

0

e^−ss^a−1 Z ∞

0

e^−z− e^−zs

z dz ds

= Z ∞

0

e^−z

Z ∞ 0

s^a−1e^−sds − Z ∞

0

s^a−1e^−s(1+z)ds

dz z . This formula can be rewritten as

Γ^′(a) = Γ(a) Z ∞

0

e^−z− (1 + z)^−a dz z .

This establishes (2.1).

Example 2.2. The special case a = 1 yields (2.6)

Z ∞ 0

e^−x− 1 1 + x

dx x = −γ.

This appears as 3.435.3.

Example 2.3. The change of variables w = − ln x gives the value of 4.275.2:

(2.7)

Z 1 0

x −

1

1 − ln x

q dx x ln x = −

Z ∞ 0

e^−w− (1 + w)^−q dw

w = −ψ(q).

Example 2.4. The change of variables t = 1/(x + 1) in (2.1) yields 3.471.14:

(2.8)

Z 1 0

e^(1−1/t)− t^a

t(1 − t) dt = ψ(a)

Example 2.5. The result of Example 2.2 can be used to prove 3.435.4:

(2.9)

Z ∞ 0

e^−bx− 1 1 + ax

dx x = lna

b − γ.

Indeed, the change of variables t = bx yields from (2.2) the identity Z ∞

0

e^−bx− 1 1 + ax

dx

x =

Z ∞ 0

e^−t− 1 1 + at/b

dt t

= Z ∞

0

e^−t− e^−at/b

t dt +

Z ∞ 0

e^−at/b− 1 1 + at/b

dt t . Formula (2.5) shows the first integral is ln^a_b and the value of the second one comes from (2.2).

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Example 2.6. The evaluation 3.476.2:

(2.10)

Z ∞ 0

e^−x^p− e^−x^qdx

x = p − q pq γ

comes directly from (2.1). Indeed, the change of variables u = x^p yields I :=

Z ∞ 0

e^−x^p− e^−x^qdx x = 1

p Z ∞

0

e^−u− e^−u^q/pdu u. Now write

I =1 p

Z ∞ 0

e^−u− 1 1 + u

du u +1

p Z ∞

0

1

1 + u− e^−u^q/p du u. The first integral is −γ by (2.6) and the change of variables v = u^q/pgives

I = −γ p+1

q Z ∞

0

1

1 + v^p/q − e^−v dv v

= −γ p+1

q Z ∞

0

1

1 + v − e^−v dv v +1

q Z ∞

0

v − v^p/q

v(1 + v)(1 + v^p/q)dv.

Split the last integral from [0, 1] to [1, ∞) and use the change of variables x 7→ 1/x in the second part to check that the whole integral vanishes. Formula (2.10) has been established.

Example 2.7. Formula 3.463:

(2.11)

Z ∞ 0

e^−x²− e^−xdx x =γ

2 corresponds to the choice p = 2 and q = 1 in (2.10).

Example 2.8. Formula 3.469.2:

(2.12)

Z ∞ 0

e^−x⁴− e^−xdx x = 3γ

(2.13)

Z ∞ 0

e^−x⁴− e^−x²dx x =γ

(2.14)

Z ∞ 0

e^−x²ⁿ− e^−xdx

x = (1 − 2⁻ⁿ)γ corresponds to the choice p = 2ⁿ and q = 1 in (2.10).

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The case p = q in (2.10) is now modified to include a parameter.

Proposition 2.11. Let a, b, p ∈ R⁺. Then 3.476.1 in [2] states that (2.15)

Z ∞ 0

he^−ax^p− e^−bx^pidx

x = ln b − ln a

p .

Proof. The change of variables t = ax^p gives Z ∞

0

he^−ax^p− e^−bx^pidx x = 1

p Z ∞

0

e^−t− e^−bt/adt t . Introduce the term 1/(1 + t) to obtain

I = 1

p Z ∞

0

e^−t− 1 1 + t

dt t −1

p Z ∞

0

e^−bt/a− 1 1 + t

dt t

= −γ p−1

p Z ∞

0

e^−s− b b + as

ds s . Adding and subtracting the term 1/(1 + s) produces

(2.16) I =1

p Z ∞

0

b

b + as− 1 1 + s

ds s .

The final result now comes from evaluating the last integral. We now present another integral representation of the digamma function.

Proposition 2.12. The digamma function is given by

(2.17) ψ(a) =

Z ∞ 0

e^−x

x − e^−ax 1 − e^−x

dx.

This expression appears as 3.427.1 in [2].

Proof. The representation (2.1) is written as

(2.18) ψ(a) = lim

δ→0

Z ∞ δ

e^−z z dz −

Z ∞ δ

dz z(1 + z)^a,

to avoid the singularity at z = 0. The change of variables z = e^t− 1 in the second integral gives

(2.19) ψ(a) = lim

δ→0

Z ∞ δ

e^−z z dz −

Z ∞ ln(1+δ)

e^−atdt 1 − e^−t. Now observe that

(2.20)

Z ln(1+δ) δ

e^−t t dt

6 Z δ

ln(1+δ)

dt t → 0,

as δ → 0. This completes the proof.

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Example 2.13. The special case a = 1 in (2.17) gives 3.427.2:

(2.21)

Z ∞ 0

1

1 − e^−x −1 x

e^−xdx = γ.

Example 2.14. The change of variables t = e^−xin (2.17) produces 4.281.4:

(2.22)

Z 1 0

1

ln t + t^a−1 1 − t

dt = −ψ(a).

Example 2.15. The special case a = 1 in (2.22) yields 4.281.1:

(2.23)

Z 1 0

1 ln t+ 1

1 − t

dt = γ.

Proposition 2.16. Let p, q ∈ R. Then (2.24)

Z 1 0

x^p−1

ln x + x^q−1 1 − x

dx = ln p − ψ(q).

This appears as 4.281.5 in [2].

Proof. Write (2.25)

Z 1 0

x^p−1

ln x + x^q−1 1 − x

dx =

Z 1 0

1

ln x+ x^q−1 1 − x

dx +

Z 1 0

x^p−1− 1 ln x dx.

The first integral is −ψ(q) from (2.22) and to evaluate the second one, differentiate with respect to p, to produce

(2.26) d

dp Z 1

0

x^p−1− 1 ln x dx =

Z 1 0

x^p−1dx = 1 p.

The value at p = 1 shows that the constant of integration vanishes. The formula (2.24)

has been established.

3. The difference of values of the digamma function

In this section we establish an integral representation for the difference of values of the digamma function. The expression appears as 3.231.5 in [2].

Z 1 0

x^p−1− x^q−1

1 − x dx = ψ(q) − ψ(p).

Proof. Consider first

(3.2) I(ǫ) =

Z 1 0

x^p−1(1 − x)^ǫ−1dx − Z 1

0

x^q−1(1 − x)^ǫ−1dx,

that avoids the apparent singularity at x = 1. The integral I(ǫ) can be expressed in terms of the beta function

(3.3) B(a, b) =

Z 1 0

x^a−1(1 − x)^b−1dx

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as I(ǫ) = B(p, ǫ) − B(q, ǫ), and using the relation

(3.4) B(a, b) = Γ(a) Γ(b)

Γ(a + b) we obtain

(3.5) I(ǫ) = Γ(ǫ)

Γ(p)

Γ(p + ǫ)− Γ(q) Γ(q + ǫ)

. Now use Γ(1 + ǫ) = ǫΓ(ǫ) to write

(3.6) I(ǫ) = Γ(1 + ǫ) Γ(p) − Γ(p + ǫ) ǫ

1

Γ(p + ǫ)−Γ(q) − Γ(q + ǫ) ǫ

1 Γ(q + ǫ)

,

and obtain (3.1) by letting ǫ → 0.

Example 3.2. The special value ψ(1) = −γ produces (3.7)

Z 1 0

1 − x^q−1

1 − x dx = γ + ψ(q).

This appears as 3.265 in [2].

Example 3.3. A second special value appears in 3.268.2:

(3.8)

Z 1 0

1 − x^a

1 − x x^b−1dx = ψ(a + b) − ψ(b).

It is obtained from (3.1) by choosing p = b and q = a + b.

Example 3.4. Now let q = 1 − p in (3.1) to produce (3.9)

Z 1 0

x^p−1− x^−p

1 − x dx = ψ(1 − p) − ψ(p) = π cot πp.

Example 3.5. The special case p = a + 1 and q = 1 − a produces (3.10)

Z 1 0

x^a− x^−a

1 − x dx = ψ(1 − a) − ψ(1 + a) = π cot πa −1 a,

where we have used (1.10) and (1.16) to simplify the result. This is 3.231.3 in [2].

Example 3.6. The change of variables x = t^a in (3.1) produces (3.11)

Z 1 0

t^ap−1− t^aq−1

1 − t^a dt = ψ(q) − ψ(p)

a .

Now let p = 1, a = ν and q = ^µ_ν and the replace µ by p and ν by q to obtain 3.244.3 in [2]:

(3.12)

Z 1 0

t^q−1− t^p−1 1 − t^q dt = 1

q

γ + ψ p q

.

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Example 3.7. The special case p = b/a and q = 1 − b/a in (3.11) produces (3.13)

Z 1 0

x^b−1− x^a−b−1

1 − x^a dx = 1

a(ψ(1 − b/a) − ψ(b/a)) . The result is now simplified using (1.16) to produce

(3.14)

Z 1 0

x^b−1− x^a−b−1

1 − x^a dx = π a cot πb

a. This is 3.244.2 in [2].

Example 3.8. The special case a = 2 in (3.11) yields (3.15)

Z 1 0

t^2µ−1− t^2ν−1 1 − t² dt = 1

2(ψ(ν) − ψ(µ)) . The choice µ = 1 + p/2 and ν = 1 − p/2:

(3.16)

Z 1 0

x^p− x^−p

1 − x² x dx = 1

2(ψ(1 + p/2) − ψ(1 − p/2)) .

The identities ψ(x + 1) = ψ(x) + 1/x and ψ(1 − x) − ψ(x) = π cot πx produce (3.17)

Z 1 0

x^p− x^−p

1 − x² x dx = π

2cotpπ 2

−1 p. This appears as 3.269.1 in [2].

Example 3.9. The choice µ =^a+1₂ and ν = ^b+1₂ in (3.15) gives 3.269.3:

(3.18)

Z 1 0

x^a− x^b 1 − x² dx = 1

2

ψ b + 1 2

− ψ a + 1 2

. 4. Integrals over a half-line

In this section we consider integrals over the half-line [0, ∞) that can be evaluated in terms of the digamma function.

Z ∞ 0

t^p

(1 + t)^p − t^q (1 + t)^q

dt

t = ψ(q) − ψ(p).

This is 3.219 in [2]. Also (4.2)

Z ∞ 0

1

(1 + t)^p − 1 (1 + t)^q

dt

t = ψ(q) − ψ(p).

Proof. Let t = x/(1 − x) in (3.1). The second form comes from the first by the

change of variables x 7→ 1/x.

Example 4.2. The special case p = 1 yields (4.3)

Z ∞ 0

1

1 + t− 1 (1 + t)^q

dt

t = ψ(q) + γ.

This appears as 3.233 in [2].

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Example 4.3. The evaluation of 3.235:

(4.4)

Z ∞ 0

(1 + x)^a− 1 (1 + x)^b

dx

x = ψ(b) − ψ(b − a) can be established directly from (4.3). Simply write

Z ∞ 0

(1 + x)^a− 1 (1 + x)^b

dx x =

Z ∞ 0

1

1 + x− 1 (1 + x)^b

dx x −

Z ∞ 0

1

1 + x− 1 (1 + x)^b−a

dx x to obtain the result.

Some examples of integrals over [0, ∞) can be reduced to a pair of integrals over [0, 1].

Proposition 4.4. The formula 3.231.6 of [2] states that (4.5)

Z ∞ 0

x^p−1− x^q−1

1 − x dx = π (cot πp − cot πq) .

Proof. To evaluate this, make the change of variables t = 1/x in the part over [1, ∞) to produce

(4.6)

Z ∞ 0

x^p−1− x^q−1 1 − x dx =

Z 1 0

x^p−1− x^q−1 1 − x dx −

Z 1 0

x^−p− x^−q 1 − x dx.

Now use the result (3.1) to write (4.7)

Z ∞ 0

x^p−1− x^q−1

1 − x dx = ψ(q) − ψ(p) − [ψ(1 − q) − ψ(1 − p)] .

The relation ψ(x) − ψ(1 − x) = −π cot(πq) yields the result. 5. An exponential scale

In this section we present the evaluation of certain definite integrals involving the exponential function. These are integrals that can be evaluated in terms of the digamma function of the parameters involved.

Example 5.1. The simplest one is 3.317.2:

(5.1)

Z ∞

−∞

1

(1 + e^−x)^p − 1 (1 + e^−x)^q

dx = ψ(q) − ψ(p) that comes from (4.2) via the change of variables x 7→ e^−x.

Example 5.2. The special case p = 1 and ψ(1) = −γ produces 3.317.1:

(5.2)

Z ∞

−∞

1

1 + e^−x − 1 (1 + e^−x)^q

dx = ψ(q) + γ Example 5.3. The evaluation of 3.316:

(5.3)

Z ∞

−∞

(1 + e^−x)^p− 1

(1 + e^−x)^q dx = ψ(q) − ψ(q − p) comes directly from (5.1).

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Z ∞ 0

e^−pt− e^−qt

1 − e^−t dt = ψ(q) − ψ(p), This appears as 3.311.7 in [2].

Proof. Make the change of variables x = e^−tin (3.1). Example 5.5. The evaluation (5.4) can also be written as

(5.5)

Z ∞ 0

e^t(1−p)− e^t(1−q)

e^t− 1 dt = ψ(q) − ψ(p), Example 5.6. The special case p = 1 and q = 1 − ν is (5.6)

Z ∞ 0

1 − e^νt

e^t− 1 dt = ψ(1 − ν) − ψ(1),

and using ψ(1) = −γ and ψ(1 − ν) = ψ(ν) + π cot πν, yields the form (5.7)

Z ∞ 0

1 − e^νt

e^t− 1 dt = ψ(ν) + γ + π cot πν, as it appears in 3.311.5.

Example 5.7. Another special case of (5.4) is 3.311.6, that corresponds to p = 1:

(5.8)

Z ∞ 0

e^−t− e^−qt

1 − e^−t dt = ψ(q) + γ.

Example 5.8. The evaluation 3.311.11:

(5.9)

Z ∞ 0

e^px− e^qx

e^rx− e^sx dx = 1 r − s

ψ r − q r − s

− ψ r − p r − s

, follows directly from (5.4) by the change of variables t = (r − s)x.

Example 5.9. The evaluation of 3.311.12:

(5.10)

Z ∞ 0

a^x− b^x

c^x− d^xdx = 1 ln c − ln d

ψ ln c − ln b ln c − ln d

− ψ ln c − ln a ln c − ln d

, is proved by simply writing the exponentials in natural base.

Example 5.10. The formula 3.311.10 had a sign error in the sixth edition of [2].

It appears as (5.11)

Z ∞ 0

e^−px− e^−qx

1 + e^−(p+q)xdx = π p + qcot

pπ p + q

. It should be

(5.12)

Z ∞ 0

e^−px− e^−qx

1 − e^−(p+q)xdx = π p + qcot

pπ p + q

. The value (5.9) yields

(5.13)

Z ∞ 0

e^−px− e^−qx

1 − e^−(p+q)xdx = 1 p + q

ψ

q p + q

− ψ

p

p + q

,

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and the trigonometric answer follows from (1.16). This has been corrected in the current edition of [2].

(5.14) Z ∞

0

(1 − e^−ax)(1 − e^−bx)e^−px

1 − e^−x dx = ψ(p + a) + ψ(p + b) − ψ(p + a + b) − ψ(p) follows directly from (3.1). Indeed, the change of variables t = e^−x gives

(5.15) I =

Z 1 0

t^p−1(1 − t^a− t^b+ t^a+b)

1 − t dt

and now split them as

(5.16) I =

Z 1 0

t^p−1− t^p+a−1 1 − t dt −

Z 1 0

t^p+b−1− t^p+a+b−1

1 − t dt

and use (3.1) to conclude.

6. A singular example The example discussed in this section is

(6.1)

Z ∞

−∞

e^−µxdx

b − e^−x = b^µ−1π cot(πµ),

that appears as 3.311.8 in [2]. In the case b > 0 this has to be modified in its presentation to avoid the singularity x = − ln b. The case b < 0 was discussed in [4].

In order to reduce the integral to a previous example, we let t = e^−x to obtain (6.2)

Z ∞

−∞

e^−µxdx b − e^−x =

Z ∞ 0

t^µ−1dt b − t . The change of variables t = by yields

(6.3)

Z ∞

−∞

e^−µxdx

b − e^−x = b^µ−1 Z ∞

0

y^µ−1dy 1 − y .

Now separate the range of integration into [0, 1] and [1, ∞). Then make the change of variables y = 1/z in the second part. This produces

(6.4)

Z ∞

−∞

e^−µxdx

b − e^−x = b^µ−1 Z 1

0

z^µ−1− z^−µ 1 − z dz.

This last integral has been evaluated as cot(πµ) in (3.9).

7. An integral with a fake parameter The example considered in this section is 3.234.1:

(7.1)

Z 1 0

x^q−1

1 − ax− x^−q a − x

dx = π a^q cot πq.

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We show that the parameter a is fake, in the sense that it can be easily scaled out of the formula. The integral is written as lim

ǫ→0I(ǫ) where I(ǫ) =

Z 1 0

x^q−1

(1 − ax)^1−ǫ − x^−q (a − x)^1−ǫ

dx

= Z 1

0

x^q−1

(1 − ax)^1−ǫdx − Z 1

0

x^−q (a − x)^1−ǫ dx.

Make the change of variables t = ax in the first integral and x = at in the second one to produce

I(ǫ) = a^−q Z a

0

t^q−1dt

(1 − t)^1−ǫ − a^−q+ǫ Z 1/a

0

t^−qdt (1 − t)^1−ǫ, and then let ǫ → 0 to produce

Z 1 0

x^q−1

1 − ax− x^−q a − x

dx = a^−q Z a

0

t^q−1dt 1 − t −

Z 1/a 0

t^−qdt 1 − t

! .

Differentiation with respect to the parameter a, shows that the expression in paren- thesis is independent of a. It is now evaluated by using a = 1 to obtain

Z 1 0

x^q−1

1 − ax − x^−q a − x

dx = a^−q

Z 1 0

t^q−1− t^−q 1 − t dt

. The evaluation (3.1) now yields

Z 1 0

x^q−1

1 − ax− x^−q a − x

dx = a^−q(ψ(1 − q) − ψ(q))

= a^−qπ cot πq.

Formula (7.1) has been established.

8. The derivative of ψ

In a future publication we will discuss the evaluation of definite integrals in terms of the polygamma function

(8.1) PolyGamma[n, x] := d

dx

n

ψ(x).

In this section, we simply describe some integrals in [2] that comes from direct differentiation of the examples described above.

Example 8.1. Differentiating (3.1) with respect to the parameter p produces 4.251.4:

(8.2)

Z 1 0

x^p−1 ln x

1 − x dx = −ψ^′(p).

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Example 8.2. The change of variables x = t^q in (8.2), followed by the change of parameter p 7→ ^p_q yields 4.254.1:

(8.3)

Z 1 0

t^p−1 ln t

1 − t^q dx = −1 q²ψ^′ p

q

.

Example 8.3. Replace q by 2q and p by q in (8.3) to produce (8.4)

Z 1 0

t^q−1 ln t

1 − t^2q dt = − 1

4q²ψ^′ ¹₂ .

To evaluate this last term, differentiate the logarithm of the identity

(8.5) Γ(2x) =2^2x−1

√π Γ(x) Γ(x +¹₂), to obtain

(8.6) 2ψ(2x) = 2 ln 2 + ψ(x) + ψ(x + ¹₂).

One more differentitation produces

(8.7) 4ψ^′(2x) = ψ^′(x) + ψ^′(x +¹₂).

The value x = ¹₂ gives

(8.8) ψ^′(¹₂) = 3ψ^′(1) = π² 2 . Therefore we obtain 4.254.6:

(8.9)

Z 1 0

x^q−1 ln x

1 − x^2q dx = −π² 8q².

Example 8.4. Differentiating (3.12) n-times with respect to the parameter p produces 4.271.15:

(8.10)

Z 1 0

lnⁿxx^p−1dx

1 − x^q = − 1

qⁿ⁺¹ψ⁽ⁿ⁾ p q

.

9. A family of logarithmic integrals

Several of the integrals appearing in [2] are particular examples of the family evaluated in the next proposition.

Proposition 9.1. Let a, b ∈ R⁺. Then (9.1)

Z 1 0

x^a−1(1 − x)^b−1ln x dx = Γ(a) Γ(b)

Γ(a + b) (ψ(a) − ψ(a + b)) . Proof. Differentiate the identity

(9.2)

Z 1 0

x^a−1(1 − x)^b−1dx = Γ(a) Γ(b) Γ(a + b)

with respect to the parameter a and recall that Γ^′(x) = ψ(x)Γ(x).

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The next corollary appears as 4.253.1 in [2].

Corollary 9.2. Let a, b, c ∈ R⁺. Then (9.3)

Z 1 0

x^a−1(1 − x^c)^b−1ln x dx = Γ(a/c) Γ(b) c²Γ(a/c + b)

ψa c

− ψa c + b

.

Proof. Let t = x^c in the integral (9.1).

Example 9.3. The formula in the previous corollary also appears as 4.256 in the form

(9.4)

Z 1 0

ln 1 x

x^µ−1dx

p(1 − xn ⁿ)^n−m = 1 n²Bµ

n,m n

ψ µ + m n

− ψµ n

.

Example 9.4. The integral (9.5)

Z 1 0

x²ⁿ ln x

√1 − x²dx = Z 1

0

x²ⁿ(1 − x²)^−1/2 ln x dx

that appears as 4.241.1 in [2], corresponds to a = 2n + 1, b = ¹₂ and c = 2 in (9.3).

Therefore (9.6)

Z 1 0

x²ⁿ ln x

√1 − x²dx = Γ(n + ¹₂) Γ(¹₂)

4Γ(n + 1) ψ(n +¹₂) − ψ(n + 1) . Using (1.7), (1.11) and (1.14) yields

(9.7)

Z 1 0

x²ⁿ ln x

√1 − x²dx =

2n n π 2²ⁿ⁺¹

2n

X

k=1

(−1)^k−1 k − ln 2

! . This is 4.241.1.

Example 9.5. The integral in 4.241.2 states that (9.8)

Z 1 0

x²ⁿ⁺¹ ln x

√1 − x² dx = (2n)!!

(2n + 1)!! ln 2 +

2n+1

X

k=1

(−1)^k k

! . Writing the integral as

(9.9) I =

Z 1 0

x²ⁿ⁺¹(1 − x²)^−1/2 ln x dx

we see that is corresponds to the case a = 2n + 2, b = ¹₂, c = 2 in (9.3). Therefore (9.10) I = Γ(n + 1) Γ(¹₂)

4Γ(n +³₂) ψ(n + 1) − ψ(n +³2) . Using (1.7), (1.11) and (1.14) yields

(9.11)

Z 1 0

x²ⁿ⁺¹ ln x

√1 − x² dx = 2²ⁿ

(n + 1) ²ⁿ⁺¹_n ln 2 +

2n+1

X

k=1

(−1)^k k

!

This is equivalent to (9.8).

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Example 9.6. The integral 4.241.3 in [2] states that (9.12)

Z 1 0

x²ⁿp

1 − x² ln x dx = (2n − 1)!!

(2n + 2)!!·π 2

2n

X

k=1

(−1)^k−1

k − 1

2n + 2− ln 2

! . To evaluate the integral, we write it as

(9.13) I =

Z 1 0

x²ⁿ(1 − x²)^1/2 ln x dx

and we see that is corresponds to the case a = 2n + 1, b = ³₂, c = 2 in (9.3). Therefore (9.14) I = Γ(n +¹₂) Γ(³₂)

4Γ(n + 2) ψ(n +¹₂) − ψ(n + 2) . Using (1.7), (1.11) and (1.14) yields

(9.15) Z 1

0

x²ⁿp

1 − x²ln x dx = −

2n n π

2²ⁿ⁺²(n + 1) ln 2 + 1 2n + 2 +

2n

X

k=1

(−1)^k k

! . This is equivalent to (9.12).

Z 1 0

x²ⁿ⁺¹p

1 − x²ln x dx = (2n)!!

(2n + 3)!! ln 2 +

2n+1

X

k=1

(−1)^k−1

k − 1

2n + 3

! . To evaluate the integral, we write it as

(9.17) I =

Z 1 0

x²ⁿ⁺¹(1 − x²)^1/2ln x dx

and we see that is corresponds to the case a = 2n + 2, b = ³₂, c = 2 in (9.3). Therefore (9.18) I = Γ(n + 1) Γ(³₂)

4Γ(n +⁵₂) ψ(n + 1) − ψ(n +⁵2) . Using (1.7), (1.11) and (1.14) yields

(9.19) Z 1

0

x²ⁿ⁺¹p

1 − x² ln x dx = 2²ⁿ⁺¹

(n + 1)(n + 2) ²ⁿ⁺³_n+1 ln 2 − 1 2n + 3 +

2n+1

X

k=1

(−1)^k k

! . This is equivalent to (9.16).

Z 1 0

ln xp(1 − x²)²ⁿ⁻¹dx = −(2n − 1)!! π

4 (2n)!! [ψ(n + 1) + γ + ln 4]

To evaluate the integral, we write it as

(9.21) I =

Z 1

0 (1 − x²)ⁿ⁻¹² ln x dx

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and we see that is corresponds to the case a = 1, b = n + ¹₂, c = 2 in (9.3). Therefore (9.22) I =Γ(n +¹₂) Γ(¹₂)

4Γ(n + 1) ψ(¹₂) − ψ(n +¹2) . Using (1.7), (1.11) and (1.14) yields

(9.23)

Z 1

0 (1 − x²)ⁿ⁻¹² ln x dx = −

2n n π

2²ⁿ⁺² 2 ln 2 +

n

X

k=1

1 k

! . This is equivalent to (9.20). This integral also appears as 4.246.

Example 9.9. The case n = 0 in (9.7) yields (9.24)

Z 1 0

ln x dx

√1 − x² = −π 2ln 2.

Example 9.10. Formula 4.241.8 states that (9.25)

Z ∞ 1

ln x dx x²√

x²− 1 = 1 − ln 2.

To evaluate this, let t = 1/x to obtain

(9.26) I = −

Z 1

0 t(1 − t²)^−1/2 ln t dt.

This corresponds to the case a = 2, b = ¹₂, c = 2 in (9.3). Therefore

(9.27) I = −Γ(1) Γ(¹₂)

4 Γ(³₂) ψ(1) − ψ(³2) , and the value 1 − ln 2 comes from (1.11) and (1.14).

Example 9.11. The case n = 0 in (9.15) produces (9.28)

Z 1 0

p1 − x² ln x dx = −π

8(2 ln 2 + 1).

Example 9.12. The case n = 0 in (9.19) produces (9.29)

Z 1 0

xp

1 − x² ln x dx = 1

9(3 ln 2 − 4).

Example 9.13. Entry 4.241.11 states that (9.30)

Z 1 0

ln x dx px(1 − x²)= −

√2π

8 Γ² ¹₄ . To evaluate the integral, write it as

(9.31) I =

Z 1 0

x^−1/2(1 − x²)^−1/2 ln x dx

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and this corresponds to the case a = ¹₂, b = ¹₂, c = 2 in (9.3). Therefore (9.32) I = Γ(¹₄) Γ(¹₂)

4Γ(³₄) ψ ¹₄ − ψ ³4 . The stated form comes from using (1.9) and (1.16).

Example 9.14. The identity (9.33)

Z 1 0

x ln x

√1 − x⁴dx = −π 8 ln 2 appears as 4.243 in [2]. To evaluate it, we write it as

(9.34) I =

Z 1

0 x(1 − x⁴)^−1/2 ln x dx that corresponds to a = 2, b = ¹₂, c = 4 in (9.3). Therefore,

(9.35) I = 1

16Γ²(¹₂)ψ(¹₂) − ψ(1) . The values ψ(1) = −γ and ψ(¹₂) = −γ − 2 ln 2 gives the result.

Example 9.15. The verification of 4.244.1:

(9.36)

Z 1 0

ln x dx

px(1 − x3 ²)² = −1 8Γ³ ¹₃ is achieved by using (9.3) with a = ²₃, b = ¹₃ and c = 2 to obtain

(9.37) I = Γ² ¹₃

4Γ(²₃) ψ ¹₃ − ψ ²3 . Using (1.9) and (1.16) produces the stated result.

Example 9.16. The usual application of (9.3) shows that 4.244.2 is (9.38)

Z 1 0

ln x dx

√3

1 − x³ = 2π 9√

3 ψ ¹₃ + γ , where we have used Γ(¹₃)Γ(²₃) = 2π/√

3. It remains to evaluate ψ(¹₃). The identity (1.16) gives

(9.39) ψ ¹₃ − ψ ²3 = − π

√3.

To obtain a second relation among these quantities, we start from the identity

(9.40) Γ(3x) =3^3x−1/2

2π Γ(x) Γ(x +¹₃) Γ(x +²₃)

that follows directly from (1.6), and differentiate logarithmically to obtain (9.41) ψ(3x) = ln 3 +1

3 ψ(x) + ψ(x +¹₃) + ψ(x +²₃ . The special case x = ¹₃ yields

(9.42) ψ ¹₃ + ψ ²₃ = −2γ − 3 ln 3.

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We conclude that

(9.43) ψ(¹₃) = −γ −3

2ln 3 − π 2√ 3 and

(9.44) ψ(²₃) = −γ −3

2ln 3 + π 2√

3. This gives

(9.45)

Z 1 0

ln x dx

√3

1 − x³ = − π 3√ 3

ln 3 + π 3√ 3

, as stated in 4.244.2.

(9.46)

Z 1 0

ln x dx

√3

1 − x³ = − π 3√ 3

ln 3 − π 3√ 3

, proceeds as in the previous example. The integral is identified as (9.47) I = ¹₉Γ(²₃)Γ(¹₃)ψ ²₃ + γ .

The value (9.44) gives the rest.

Example 9.18. The change of variables t = x⁴ yields (9.48)

Z 1 0

x^p ln x dx

√1 − x⁴ = 1 16

Z 1 0

t^(p−3)/4(1 − t)^−1/2 ln t dt.

The last integral is evaluated using (9.3) with a = ^p+1₄ , b = ¹₂ and c = 1 to obtain (9.49)

Z 1 0

x^p ln x dx

√1 − x⁴ =

√π 16

Γ(^p+1₄ ) Γ(^p+3₄ )

ψ p + 1 4

− ψ p + 3 4

. The special case p = 4n + 1 yields

(9.50)

Z 1 0

x⁴ⁿ⁺¹ ln x dx

√1 − x⁴ =

√π

16n!Γ(n + ¹₂)ψ(n +¹₂) − ψ(n + 1) . The special case p = 4n + 1 yields

Z 1 0

√1 − x⁴ =

√π Γ(n +¹₂)

16n! ψ(n +¹₂) − ψ(n + 1) . Using (1.7), (1.11) and (1.14) yields 4.245.1 in the form

(9.51)

Z 1 0

√1 − x⁴ =π ²ⁿ_n

2²ⁿ⁺³ − ln 2 +

2n

X

k=1

(−1)^k−1 k

! . The special case p = 4n + 3 yields

Z 1 0

x⁴ⁿ⁺³ ln x dx

√1 − x⁴ =

√π n!

16Γ(n +³₂)ψ(n + 1) − ψ(n +³2) .

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Using (1.7), (1.11) and (1.14) yields 4.245.2 in the form (9.52)

Z 1 0

x⁴ⁿ⁺³ ln x dx

√1 − x⁴ = 2²ⁿ⁻²

(2n + 1) ²ⁿ_n ln 2 +

2n+1

X

k=1

(−1)^k k

! .

Example 9.19. The change of variables t = x²ⁿ produces (9.53)

Z 1 0

ln x dx

√n

1 − x²ⁿ = 1 4n²

Z 1 0

t²ⁿ¹ ⁻¹(1 − t)⁻ⁿ¹ ln t dt.

Then (9.3) with a = _2n¹, b = 1 −_n¹ and c = 1 give the value (9.54)

Z 1 0

ln x dx

√n

1 − x²ⁿ = Γ(_2n¹) Γ(1 −_n¹)

4n²Γ(1 −2n¹ ) ψ _2n¹ − ψ 1 −_2n¹ . Using (1.7) and (1.14) to obtain

(9.55)

Z 1 0

ln x dx

√n

1 − x²ⁿ = −π 8

B _2n¹,_2n¹ n² sin _2n^π . This is 4.247.1 in [2].

Example 9.20. The change of variables t = x² gives Z 1

0

ln x dx

pxn ⁿ⁻¹(1 − x²) = 1 4

Z 1 0

t²ⁿ¹⁻¹(1 − t)⁻ⁿ¹ ln t dt.

Using (9.3) we obtain (9.56)

Z 1 0

ln x dx

pxn ⁿ⁻¹(1 − x²) =Γ(_2n¹) Γ(1 − _2n¹ )

Γ(1 −_2n¹) ψ _2n¹ − ψ 1 − 2n¹ . Proceeding as in the previous example, we obtain

(9.57)

Z 1 0

ln x dx

pxn ⁿ⁻¹(1 − x²)= −π 8

B _2n¹ ,_2n¹ sin _2n^π . This is 4.247.2 in [2].

Some integrals in [2] have the form of the Corollary 9.2 after an elementary change of variables.

Example 9.21. Formula 4.293.8 in [2] states that (9.58)

Z 1 0

x^a−1ln(1 − x) dx = −1

a(ψ(a + 1) + γ) .

This follows directly from (9.3) by the change of variables x 7→ 1 − x. The same is true for 4.293.13:

(9.59)

Z 1 0

x^a−1(1 − x)^b−1ln(1 − x) dx = B(a, b) [ψ(b) − ψ(a + b)] .

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Example 9.22. The change of variables t = e^−xgives (9.60)

Z ∞ 0

xe^−x(1 − e^2x)ⁿ⁻¹²dx = − Z 1

0 (1 − t²)ⁿ⁻¹² ln t dt.

This latter integral is evaluated using (9.3) as

(9.61) I = −

√πΓ(n + ¹₂)

4n! ψ ¹₂ − ψ(n + 1) . Using (1.7) and (1.11) we obtain

(9.62)

Z ∞ 0

xe^−x(1 − e^−2x)ⁿ⁻¹²dx =

2n n π

2²ⁿ⁺² 2 ln 2 +

n

X

k=1

1 k

! . This appears as 3.457.1 in [2].

10. An announcement

There are many integrals in [2] that contain the term 1 + x in the denominator, instead of the term 1 − x seen, for instance, in Section 3. The evaluation of these integrals can be obtained using the incomplete beta function, defined by

(10.1) β(a) :=

Z 1 0

x^a−1dx 1 + x

as it appears in 8.371.2. This function is related to the digamma function by the identity

(10.2) β(a) = 1

2

ψ a + 1 2

− ψa 2

. These evaluations will be reported in [5].

11. One more family

We conclude this collection with a two-parameter family of integrals.

Proposition 11.1. Let a, b ∈ R⁺. Then (11.1)

Z ∞ 0

e^−x^a− 1 1 + x^b

dx x = −γ

a, independently of b.

Proof. Write the integral as (11.2)

Z ∞ 0

e^−x^a− e^−x^b dx x +

Z ∞ 0

e^−x^b− 1 1 + x^b

dx x.

The first integral is (a − b)γ/ab according to (2.10). The change of variables t = x^b converts the second one into

(11.3) 1

b Z ∞

0

e^−t− 1 1 + t

dt t = −γ

b,

according to (2.6). The formula has been established.

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Example 11.2. The case a = 2ⁿ and b = 2ⁿ⁺¹gives 3.475.1:

(11.4)

Z ∞ 0

exp(−x²ⁿ) − 1 1 + x²ⁿ⁺¹

dx x = −γ

2ⁿ. Example 11.3. The case a = 2ⁿ and b = 2 gives 3.475.2:

(11.5)

Z ∞ 0

exp(−x²ⁿ) − 1 1 + x²

dx x = −γ

2ⁿ. Example 11.4. The case a = 2 and b = 2 gives 3.467:

(11.6)

Z ∞ 0

e^−x²− 1 1 + x²

dx x = −γ

2. Example 11.5. Finally, the change of variables t = ax yields (11.7)

Z ∞ 0

e^−px− 1 1 + a²x²

dx x =

Z ∞ 0

e^−pt/a− e^−t

t dt +

Z ∞ 0

e^−t− 1 1 + t²

dt t . The first integral is ln^a_p according to (2.5), the second one is −γ. This gives the evaluation of 3.442.3:

(11.8)

Z ∞ 0

e^−px− 1 1 + a²x²

dx

x = γ + lna p.

References

[1] G. Boros and V. Moll. Irresistible Integrals. Cambridge University Press, New York, 1st edition, 2004.

[2] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition, 2007.

[3] V. Moll. The integrals in Gradshteyn and Ryzhik. Part 4: The gamma function. Scientia, 15:37–

46, 2007.

[4] V. Moll. The integrals in Gradshteyn and Ryzhik. Part 6: The beta function. Scientia, 16:9–24, 2008.

[5] V. Moll. The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function. Scientia, to appear.

Department of Mathematics, Tulane University, New Orleans, LA 70118 E-mail address: lmedina@math.tulane.edu

Department of Mathematics, Tulane University, New Orleans, LA 70118 E-mail address: vhm@math.tulane.edu

Received 29 11 2007, revised 18 11 2008