The integrals in Gradshteyn and Ryzhik. Part 10: the digamma function
aa r X i v : . [ m a t h . C A ] S e p 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 thatcan be expressed in terms of the digamma function ψ ( x ) = ddx log Γ( x ). Inthis note we present some of these evaluations. Introduction
The table of integrals [2] contains a large variety of definite integrals that involvethe digamma function(1.1) ψ ( x ) = ddx log Γ( x ) = Γ ′ ( x )Γ( x ) . Here Γ( x ) is the gamma function defined by(1.2) Γ( x ) = Z ∞ t x − e − t dt. Many of the analytic properties can be derived from those of Γ( x ). The nexttheorem represents a collection of the important properties of Γ( x ) that are usedin 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) = x Γ( x ) . b) For n ∈ N , the interpolation formula Γ( n ) = ( n − Euler constant γ , defined by(1.4) γ = lim n →∞ n X k =1 k − ln n, is also given by γ = − Γ ′ (1). This appears as the special case a = 1 of formula . . :(1.5) Z ∞ e − ax ln x dx = − γ + ln aa . 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 . 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 firstauthor was partially supported as a graduate student by the same grant. d) The infinite product representation(1.6) Γ( x ) = e − γx x ∞ Y k =1 (cid:20)(cid:16) xk (cid:17) − e x/k (cid:21) is valid for x ∈ C away from the poles at x = 0 , − , − , . . . e) For n ∈ N we have(1.7) Γ (cid:0) n + (cid:1) = (2 n )!2 n n ! √ π and(1.8) Γ (cid:0) − n (cid:1) = ( − n n n !(2 n )! √ π. f) For x ∈ C , x 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 gammafunction. Theorem 1.2.
The digamma function ψ ( x ) satisfiesa) the functional equation(1.10) ψ ( x + 1) = ψ ( x ) + 1 x . b) For n ∈ N , we have(1.11) ψ ( n ) = − γ + n − X k =1 k . In particular, ψ (1) = − γ .c) For x ∈ C away from x = 0 , − , − , · · · we have ψ ( x ) = − γ − x + x ∞ X k =1 k ( x + k )(1.12) = − γ − ∞ X k =0 (cid:18) x + k − k + 1 (cid:19) d) The derivative of ψ is given by(1.13) ψ ′ ( x ) = ∞ X k =0 x + k ) . In particular, ψ ′ (1) = π / n ∈ N we have(1.14) ψ ( ± n ) = − γ − n X k =1 k − . In particular,(1.15) ψ ( ) = − γ − . IGAMMA FUNCTION 3 f) For x ∈ C , x Z we have the reflection rule(1.16) ψ (1 − x ) = ψ ( x ) + π cot πx. A first integral representation
In this section we establish the integral evaluation . . Severeal direct conse-quences of this formulas are also described. Proposition 2.1.
Assume a >
0. Then(2.1) Z ∞ (cid:2) e − x − (1 + x ) − a (cid:3) dxx = ψ ( a ) . Proof.
We begin with the double integral(2.2) Z ∞ Z s e − tz dt dz = Z ∞ e − z − e − sz z dz. On the other hand,(2.3) Z s Z ∞ e − tz dz dt = Z s dtt = ln s. We conclude that(2.4) Z ∞ e − z − e − sz z dz = ln s. This evaluation is equivalent to:(2.5) Z ∞ e − ax − e − bx x dx = ln ba , that appears as formula . . in [2]. The reader will find a proof in [3].We now establish the result: start withΓ ′ ( a ) = Z ∞ e − s s a − ln s ds = Z ∞ e − s s a − Z ∞ e − z − e − zs z dz ds = Z ∞ (cid:18) e − z Z ∞ s a − e − s ds − Z ∞ s a − e − s (1+ z ) ds (cid:19) dzz . This formula can be rewritten asΓ ′ ( a ) = Γ( a ) Z ∞ (cid:0) e − z − (1 + z ) − a (cid:1) dzz . This establishes (2.1). (cid:3)
Example 2.2.
The special case a = 1 yields(2.6) Z ∞ (cid:18) e − x −
11 + x (cid:19) dxx = − γ. This appears as . . . Example 2.3.
The change of variables w = − ln x gives the value of . . :(2.7) Z (cid:20) x − (cid:18) − ln x (cid:19) q (cid:21) dxx ln x = − Z ∞ (cid:2) e − w − (1 + w ) − q (cid:3) dww = − ψ ( q ) . L. MEDINA AND V. MOLL
Example 2.4.
The change of variables t = 1 / ( x + 1) in (2.1) yields . . :(2.8) Z e (1 − /t ) − t a t (1 − t ) dt = ψ ( a ) Example 2.5.
The result of Example 2.2 can be used to prove . . :(2.9) Z ∞ (cid:18) e − bx −
11 + ax (cid:19) dxx = ln ab − γ. Indeed, the change of variables t = bx yields from (2.2) the identity Z ∞ (cid:18) e − bx −
11 + ax (cid:19) dxx = Z ∞ (cid:18) e − t −
11 + at/b (cid:19) dtt = Z ∞ e − t − e − at/b t dt + Z ∞ (cid:18) e − at/b −
11 + at/b (cid:19) dtt .
Formula (2.5) shows the first integral is ln ab and the value of the second one comesfrom (2.2). Example 2.6.
The evaluation . . :(2.10) Z ∞ (cid:16) e − x p − e − x q (cid:17) dxx = p − qpq γ comes directly from (2.1). Indeed, the change of variables u = x p yields I := Z ∞ (cid:16) e − x p − e − x q (cid:17) dxx = 1 p Z ∞ (cid:16) e − u − e − u q/p (cid:17) duu . Now write I = 1 p Z ∞ (cid:18) e − u −
11 + u (cid:19) duu + 1 p Z ∞ (cid:18)
11 + u − e − u q/p (cid:19) duu . The first integral is − γ by (2.6) and the change of variables v = u q/p gives I = − γp + 1 q Z ∞ (cid:18)
11 + v p/q − e − v (cid:19) dvv = − γp + 1 q Z ∞ (cid:18)
11 + v − e − v (cid:19) dvv + 1 q Z ∞ 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 /x in the second part to check that the whole integral vanishes. Formula (2.10) hasbeen established. Example 2.7.
Formula . :(2.11) Z ∞ (cid:16) e − x − e − x (cid:17) dxx = γ p = 2 and q = 1 in (2.10). IGAMMA FUNCTION 5
Example 2.8.
Formula . . :(2.12) Z ∞ (cid:16) e − x − e − x (cid:17) dxx = 3 γ p = 4 and q = 1 in (2.10). Example 2.9.
Formula . . :(2.13) Z ∞ (cid:16) e − x − e − x (cid:17) dxx = γ p = 4 and q = 2 in (2.10). Example 2.10.
Formula . . :(2.14) Z ∞ (cid:16) e − x n − e − x (cid:17) dxx = (1 − − n ) γ corresponds to the choice p = 2 n and q = 1 in (2.10).The case p = q in (2.10) is now modified to include a paramter. Proposition 2.11.
Let a, b, p ∈ R + . Then . . in [2] states that(2.15) Z ∞ h e − ax p − e − bx p i dxx = ln b − ln ap . Proof.
The change of variables t = ax p gives Z ∞ h e − ax p − e − bx p i dxx = 1 p Z ∞ (cid:16) e − t − e − bt/a (cid:17) dtt . Introduce the term 1 / (1 + t ) to obtain I = 1 p Z ∞ (cid:18) e − t −
11 + t (cid:19) dtt − p Z ∞ (cid:18) e − bt/a −
11 + t (cid:19) dtt = − γp − p Z ∞ (cid:18) e − s − bb + as (cid:19) dss . Adding and subtracting the term 1 / (1 + s ) produces(2.16) I = 1 p Z ∞ (cid:18) bb + as −
11 + s (cid:19) dss . The final result now comes from evaluating the last integral. (cid:3)
We now present another integral representation of the digamma function.
Proposition 2.12.
The digamma function is given by(2.17) ψ ( a ) = Z ∞ (cid:18) e − x x − e − ax − e − x (cid:19) dx. This expression appears as . . in [2]. L. MEDINA AND V. MOLL
Proof.
The representation (2.1) is written as(2.18) ψ ( a ) = lim δ → Z ∞ δ e − z z dz − Z ∞ δ dzz (1 + z ) a , to avoid the singularity at z = 0. The change of variables z = e t − ψ ( a ) = lim δ → Z ∞ δ e − z z dz − Z ∞ ln(1+ δ ) e − at dt − e − t . Now observe that(2.20) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ln(1+ δ ) δ e − t t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z δ ln(1+ δ ) dtt → , as δ →
0. This completes the proof. (cid:3)
Example 2.13.
The special case a = 1 in (2.17) gives . . :(2.21) Z ∞ (cid:18) − e − x − x (cid:19) e − x dx = γ. Example 2.14.
The change of variables t = e − x in (2.17) produces . . :(2.22) Z (cid:18) t + t a − − t (cid:19) dt = − ψ ( a ) . Example 2.15.
The special case a = 1 in (2.22) yields . . :(2.23) Z (cid:18) t + 11 − t (cid:19) dt = γ. Proposition 2.16.
Let p, q ∈ R . Then(2.24) Z (cid:18) x p − ln x + x q − − x (cid:19) dx = ln p − ψ ( q ) . This appears as . . in [2]. Proof.
Write(2.25) Z (cid:18) x p − ln x + x q − − x (cid:19) dx = Z (cid:18) x + x q − − x (cid:19) dx + Z x p − − x dx. The first integral is − ψ ( q ) from (2.22) and to evaluate the second one, differentiatewith respect to p , to produce(2.26) ddp Z x p − − x dx = Z x p − dx = 1 p . The value at p = 1 shows that the constant of integration vanishes. The formula(2.24) has been established. (cid:3) IGAMMA FUNCTION 7 The difference of values of the digamma function
In this section we establish an integral representation for the difference of valuesof the digamma function. The expression appears as . . in [2]. Proposition 3.1.
Let p, q ∈ R . Then(3.1) Z x p − − x q − − x dx = ψ ( q ) − ψ ( p ) . Proof.
Consider first(3.2) I ( ǫ ) = Z x p − (1 − x ) ǫ − dx − Z x q − (1 − x ) ǫ − dx, that avoids the apparent singularity at x = 1. The integral I ( ǫ ) can be expressedin terms of the beta function(3.3) B ( a, b ) = Z x a − (1 − x ) b − dx 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 ) (cid:18) Γ( p )Γ( p + ǫ ) − Γ( q )Γ( q + ǫ ) (cid:19) . Now use Γ(1 + ǫ ) = ǫ Γ( ǫ ) to write(3.6) I ( ǫ ) = Γ(1 + ǫ ) (cid:18) Γ( p ) − Γ( p + ǫ ) ǫ p + ǫ ) − Γ( q ) − Γ( q + ǫ ) ǫ q + ǫ ) (cid:19) , and obtain (3.1) by letting ǫ → (cid:3) Example 3.2.
The special value ψ (1) = − γ produces(3.7) Z − x q − − x dx = γ + ψ ( q ) . This appears as . in [2]. Example 3.3.
A second special value appears in . . :(3.8) Z − x a − x x b − dx = 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 x p − − x − p − x dx = ψ (1 − p ) − ψ ( p ) = π cot πp. This appears as . . in [2]. L. MEDINA AND V. MOLL
Example 3.5.
The special case p = a + 1 and q = 1 − a produces(3.10) Z x a − x − a − x dx = ψ (1 − a ) − ψ (1 + a ) = π cot πa − a , where we have used (1.10) and (1.16) to simplify the result. This is . . in [2]. Example 3.6.
The change of variables x = t a in (3.1) produces(3.11) Z t ap − − t aq − − t a dt = ψ ( q ) − ψ ( p ) a . Now let p = 1 , a = ν and q = µν and the replace µ by p and ν by q to obtain . . in [2]:(3.12) Z t q − − t p − − t q dt = 1 q (cid:18) γ + ψ (cid:18) pq (cid:19)(cid:19) . Example 3.7.
The special case p = b/a and q = 1 − b/a in (3.11) produces(3.13) Z x b − − x a − b − − 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 x b − − x a − b − − x a dx = πa cot πba . This is . . in [2]. Example 3.8.
The special case a = 2 in (3.11) yields(3.15) Z t µ − − t ν − − t dt = 12 ( ψ ( ν ) − ψ ( µ )) . The choice µ = 1 + p/ ν = 1 − p/ Z x p − x − p − x x dx = 12 ( ψ (1 + p/ − ψ (1 − p/ . The identities ψ ( x + 1) = ψ ( x ) + 1 /x and ψ (1 − x ) − ψ ( x ) = π cot πx produce(3.17) Z x p − x − p − x x dx = π (cid:16) pπ (cid:17) − p . This appears as . . in [2]. Example 3.9.
The choice µ = a +12 and ν = b +12 in (3.15) gives . . :(3.18) Z x a − x b − x dx = 12 (cid:18) ψ (cid:18) b + 12 (cid:19) − ψ (cid:18) a + 12 (cid:19)(cid:19) . Integrals over a half-line
In this section we consider integrals over the half-line [0 , ∞ ) that can be evaluatedin terms of the digamma function. IGAMMA FUNCTION 9
Proposition 4.1.
Let p, q ∈ R . Then(4.1) Z ∞ (cid:18) t p (1 + t ) p − t q (1 + t ) q (cid:19) dtt = ψ ( q ) − ψ ( p ) . This is . in [2]. Also(4.2) Z ∞ (cid:18) t ) p − t ) q (cid:19) dtt = ψ ( q ) − ψ ( p ) . Proof.
Let t = x/ (1 − x ) in (3.1). The second form comes from the first by thechange of variables x /x . (cid:3) Example 4.2.
The special case p = 1 yields(4.3) Z ∞ (cid:18)
11 + t − t ) q (cid:19) dtt = ψ ( q ) + γ. This appears as . in [2]. Example 4.3.
The evaluation of . :(4.4) Z ∞ (1 + x ) a − x ) b dxx = ψ ( b ) − ψ ( b − a )can be established directly from (4.3). Simply write Z ∞ (1 + x ) a − x ) b dxx = Z ∞ (cid:18)
11 + x − x ) b (cid:19) dxx − Z ∞ (cid:18)
11 + x − x ) b − a (cid:19) dxx to obtain the result.Some examples of integrals over [0 , ∞ ) can be reduced to a pair of integrals over[0 , Proposition 4.4.
The formula . . of [2] states that(4.5) Z ∞ x p − − x q − − 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 ∞ x p − − x q − − x dx = Z x p − − x q − − x dx − Z x − p − x − q − x dx. Now use the result (3.1) to write(4.7) Z ∞ x p − − x q − − 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. (cid:3) An exponential scale
In this section we present the evaluation of certain definite integrals involvingthe exponential function. These are integrals that can be evaluated in terms of thedigamma function of the parameters involved.
Example 5.1.
The simplest one is . . :(5.1) Z ∞−∞ (cid:18) e − x ) p − e − x ) q (cid:19) dx = ψ ( q ) − ψ ( p )that comes from (4.2) via the change of variables x e − x . Example 5.2.
The special case p = 1 and ψ (1) = − γ produces . . :(5.2) Z ∞−∞ (cid:18)
11 + e − x − e − x ) q (cid:19) dx = ψ ( q ) + γ Example 5.3.
The evaluation of . :(5.3) Z ∞−∞ (1 + e − x ) p − e − x ) q dx = ψ ( q ) − ψ ( q − p )comes directly from (5.1). Proposition 5.4.
Let p, q ∈ R . Then(5.4) Z ∞ e − pt − e − qt − e − t dt = ψ ( q ) − ψ ( p ) , This appears as . . in [2]. Proof.
Make the change of variables x = e − t in (3.1). (cid:3) Example 5.5.
The evaluation (5.4) can also be written as(5.5) Z ∞ e t (1 − p ) − e t (1 − q ) e t − dt = ψ ( q ) − ψ ( p ) , Example 5.6.
The special case p = 1 and q = 1 − ν is(5.6) Z ∞ − e νt e t − 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 ∞ − e νt e t − dt = ψ ( ν ) + γ + π cot πν, as it appears in . . . Example 5.7.
Another special case of (5.4) is . . , that corresponds to p = 1:(5.8) Z ∞ e − t − e − qt − e − t dt = ψ ( q ) + γ. Example 5.8.
The evaluation . . :(5.9) Z ∞ e px − e qx e rx − e sx dx = 1 r − s (cid:18) ψ (cid:18) r − qr − s (cid:19) − ψ (cid:18) r − pr − s (cid:19)(cid:19) , follows directly from (5.4) by the change of variables t = ( r − s ) x . Example 5.9.
The evaluation of . . :(5.10) Z ∞ a x − b x c x − d x dx = 1ln c − ln d (cid:18) ψ (cid:18) ln c − ln b ln c − ln d (cid:19) − ψ (cid:18) ln c − ln a ln c − ln d (cid:19)(cid:19) , is proved by simply writing the exponentials in natural base. IGAMMA FUNCTION 11
Example 5.10.
The formula . . had a sign error in the sixth edition of [2]: it appears as(5.11) Z ∞ e − px − e − qx e − ( p + q ) x dx = πp + q cot (cid:18) pπp + q (cid:19) . It should be(5.12) Z ∞ e − px − e − qx − e − ( p + q ) x dx = πp + q cot (cid:18) pπp + q (cid:19) . The value (5.9) yields(5.13) Z ∞ e − px − e − qx − e − ( p + q ) x dx = 1 p + q (cid:18) ψ (cid:18) qp + q (cid:19) − ψ (cid:18) pp + q (cid:19)(cid:19) , and the trigonometric answer follows from (1.16). This has been corrected in thecurrent edition of [2]. Example 5.11.
The evaluation of . . :(5.14) Z ∞ (1 − e − ax )(1 − e − bx ) e − px − 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 t p − (1 − t a − t b + t a + b )1 − t dt and now split them as(5.16) I = Z t p − − t p + a − − t dt − Z t p + b − − t p + a + b − − t dt and use (3.1) to conclude. 6. A singular example
The example discussed in this section is(6.1) Z ∞−∞ e − µx dxb − e − x = b µ − π cot( πµ ) , that appears as . . in [2]. In the case b > x = − ln b . The case b < t = e − x to obtain(6.2) Z ∞−∞ e − µx dxb − e − x = Z ∞ t µ − dtb − t . The change of variables t = by yields(6.3) Z ∞−∞ e − µx dxb − e − x = b µ − Z ∞ y µ − dy − y . Now separate the range of integration into [0 ,
1] and [1 , ∞ ). Then make the changeof variables y = 1 /z in the second part. This produces(6.4) Z ∞−∞ e − µx dxb − e − x = b µ − Z z µ − − z − µ − z dz. This last integral has been evaluated as cot( πµ ) in (3.9). An integral with a fake parameter
The example considered in this section is . . :(7.1) Z (cid:18) x q − − ax − x − q a − x (cid:19) dx = πa q cot πq. We show that the parameter a is fake , in the sense that it can be easily scaled outof the formula. The integral is written as lim ǫ → I ( ǫ ) where I ( ǫ ) = Z (cid:18) x q − (1 − ax ) − ǫ − x − q ( a − x ) − ǫ (cid:19) dx = Z x q − (1 − ax ) − ǫ dx − Z x − q ( a − x ) − ǫ dx. Make the change of variables t = ax in the first integral and x = at in the secondone to produce I ( ǫ ) = a − q Z a t q − dt (1 − t ) − ǫ − a − q + ǫ Z /a t − q dt (1 − t ) − ǫ , and then let ǫ → Z (cid:18) x q − − ax − x − q a − x (cid:19) dx = a − q Z a t q − dt − t − Z /a t − q dt − 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 (cid:18) x q − − ax − x − q a − x (cid:19) dx = a − q (cid:18)Z t q − − t − q − t dt (cid:19) . The evaluation (3.1) now yields Z (cid:18) x q − − ax − x − q a − x (cid:19) dx = a − q ( ψ (1 − q ) − ψ ( q ))= a − q π 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 termsof the polygamma function (8.1) PolyGamma[ n, x ] := (cid:18) ddx (cid:19) n ψ ( x ) . In this section, we simply describe some integrals in [2] that comes from directdifferentiation of the examples described above.
Example 8.1.
Differentiating (3.1) with respect to the parameter p produces . . :(8.2) Z x p − ln x − x dx = − ψ ′ ( p ) . IGAMMA FUNCTION 13
Example 8.2.
The change of variables x = t q in (8.2), followed by the change ofparameter p pq yields . . :(8.3) Z t p − ln t − t q dx = − q ψ ′ (cid:18) pq (cid:19) . Example 8.3.
Replace q by 2 q and p by q in (8.3) to produce(8.4) Z t q − ln t − t q dt = − q ψ ′ (cid:0) (cid:1) . To evaluate this last term, differentiate the logarithm of the identity(8.5) Γ(2 x ) = 2 x − √ π Γ( x ) Γ( x + ) , to obtain(8.6) 2 ψ (2 x ) = 2 ln 2 + ψ ( x ) + ψ ( x + ) . One more differentitation produces(8.7) 4 ψ ′ (2 x ) = ψ ′ ( x ) + ψ ′ ( x + ) . The value x = gives(8.8) ψ ′ ( ) = 3 ψ ′ (1) = π . Therefore we obtain . . :(8.9) Z x q − ln x − x q dx = − π q . Example 8.4.
Differentiating (3.12) n -times with respect to the parameter p pro-duces . . :(8.10) Z ln n x x p − dx − x q = − q n +1 ψ ( n ) (cid:18) pq (cid:19) . A family of logarithmic integrals
Several of the integrals appearing in [2] are particular examples of the familyevaluated in the next proposition.
Proposition 9.1.
Let a, b ∈ R + . Then(9.1) Z x a − (1 − x ) b − ln x dx = Γ( a ) Γ( b )Γ( a + b ) ( ψ ( a ) − ψ ( a + b )) . Proof.
Differentiate the identity(9.2) Z x a − (1 − x ) b − dx = Γ( a ) Γ( b )Γ( a + b )with respect to the parameter a and recall that Γ ′ ( x ) = ψ ( x )Γ( x ). (cid:3) The next corollary appears as . . in [2]. Corollary 9.2.
Let a, b, c ∈ R + . Then(9.3) Z x a − (1 − x c ) b − ln x dx = Γ( a/c ) Γ( b ) c Γ( a/c + b ) (cid:16) ψ (cid:16) ac (cid:17) − ψ (cid:16) ac + b (cid:17)(cid:17) . Proof.
Let t = x c in the integral (9.1). (cid:3) Example 9.3.
The formula in the previous corollary also appears as . in theform(9.4) Z ln (cid:18) x (cid:19) x µ − dx n p (1 − x n ) n − m = 1 n B (cid:16) µn , mn (cid:17) (cid:20) ψ (cid:18) µ + mn (cid:19) − ψ (cid:16) µn (cid:17)(cid:21) . Example 9.4.
The integral(9.5) Z x n ln x √ − x dx = Z x n (1 − x ) − / ln x dx that appears as . . in [2], corresponds to a = 2 n + 1 , b = and c = 2 in (9.3).Therefore(9.6) Z x n ln x √ − x dx = Γ( n + ) Γ( )4Γ( n + 1) (cid:2) ψ ( n + ) − ψ ( n + 1) (cid:3) . Using (1.7), (1.11) and (1.14) yields(9.7) Z x n ln x √ − x dx = (cid:0) nn (cid:1) π n +1 n X k =1 ( − k − k − ln 2 ! . This is . . . Example 9.5.
The integral in . . states that(9.8) Z x n +1 ln x √ − x dx = (2 n )!!(2 n + 1)!! ln 2 + n +1 X k =1 ( − k k ! . Writing the integral as(9.9) I = Z x n +1 (1 − x ) − / ln x dx we see that is corresponds to the case a = 2 n + 2 , b = , c = 2 in (9.3). Therefore(9.10) I = Γ( n + 1) Γ( )4Γ( n + ) (cid:2) ψ ( n + 1) − ψ ( n + ) (cid:3) . Using (1.7), (1.11) and (1.14) yields(9.11) Z x n +1 ln x √ − x dx = 2 n ( n + 1) (cid:0) n +1 n (cid:1) ln 2 + n +1 X k =1 ( − k k ! This is equivalent to (9.8).
Example 9.6.
The integral . . in [2] states that(9.12) Z x n p − x ln x dx = (2 n − n + 2)!! · π n X k =1 ( − k − k − n + 2 − ln 2 ! . IGAMMA FUNCTION 15
To evaluate the integral, we write it as(9.13) I = Z x n (1 − x ) / ln x dx and we see that is corresponds to the case a = 2 n + 1 , b = , c = 2 in (9.3).Therefore(9.14) I = Γ( n + ) Γ( )4Γ( n + 2) (cid:2) ψ ( n + ) − ψ ( n + 2) (cid:3) . Using (1.7), (1.11) and (1.14) yields(9.15) Z x n p − x ln x dx = − (cid:0) nn (cid:1) π n +2 ( n + 1) ln 2 + 12 n + 2 + n X k =1 ( − k k ! . This is equivalent to (9.12).
Example 9.7.
The integral . . in [2] states that(9.16) Z x n +1 p − x ln x dx = (2 n )!!(2 n + 3)!! ln 2 + n +1 X k =1 ( − k − k − n + 3 ! . To evaluate the integral, we write it as(9.17) I = Z x n +1 (1 − x ) / ln x dx and we see that is corresponds to the case a = 2 n + 2 , b = , c = 2 in (9.3).Therefore(9.18) I = Γ( n + 1) Γ( )4Γ( n + ) (cid:2) ψ ( n + 1) − ψ ( n + ) (cid:3) . Using (1.7), (1.11) and (1.14) yields(9.19) Z x n +1 p − x ln x dx = 2 n +1 ( n + 1)( n + 2) (cid:0) n +3 n +1 (cid:1) ln 2 − n + 3 + n +1 X k =1 ( − k k ! . This is equivalent to (9.16).
Example 9.8.
The integral . . in [2] states that(9.20) Z ln x p (1 − x ) n − dx = − (2 n − π n )!! [ ψ ( n + 1) + γ + ln 4]To evaluate the integral, we write it as(9.21) I = Z (1 − x ) n − ln x dx and we see that is corresponds to the case a = 1 , b = n + , c = 2 in (9.3). Therefore(9.22) I = Γ( n + ) Γ( )4Γ( n + 1) (cid:2) ψ ( ) − ψ ( n + ) (cid:3) . Using (1.7), (1.11) and (1.14) yields(9.23) Z (1 − x ) n − ln x dx = − (cid:0) nn (cid:1) π n +2 n X k =1 k ! . This is equivalent to (9.20). This integral also appears as . . Example 9.9.
The case n = 0 in (9.7) yields(9.24) Z ln x dx √ − x = − π . This appears as . . in [2]. Example 9.10.
Formula . . states that(9.25) Z ∞ ln x dxx √ x − − ln 2 . To evaluate this, let t = 1 /x to obtain(9.26) I = − Z t (1 − t ) − / ln t dt. This corresponds to the case a = 2 , b = , c = 2 in (9.3). Therefore(9.27) I = − Γ(1) Γ( )4 Γ( ) (cid:2) ψ (1) − ψ ( ) (cid:3) , 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 p − x ln x dx = − π . This appears as . . in [2]. Example 9.12.
The case n = 0 in (9.19) produces(9.29) Z x p − x ln x dx = 19 (3 ln 2 − . This appears as . . in [2]. Example 9.13.
Entry . . states that(9.30) Z ln x dx p x (1 − x ) = − √ π (cid:0) (cid:1) . To evaluate the integral, write it as(9.31) I = Z x − / (1 − x ) − / ln x dx and this corresponds to the case a = , b = , c = 2 in (9.3). Therefore(9.32) I = Γ( ) Γ( )4Γ( ) (cid:2) ψ (cid:0) (cid:1) − ψ (cid:0) (cid:1)(cid:3) . The stated form comes from using (1.9) and (1.16).
IGAMMA FUNCTION 17
Example 9.14.
The identity(9.33) Z x ln x √ − x dx = − π . in [2]. To evaluate it, we write it as(9.34) I = Z x (1 − x ) − / ln x dx that corresponds to a = 2 , b = , c = 4 in (9.3). Therefore,(9.35) I = 116 Γ ( ) (cid:2) ψ ( ) − ψ (1) (cid:3) . The values ψ (1) = − γ and ψ ( ) = − γ − Example 9.15.
The verification of . . :(9.36) Z ln x dx p x (1 − x ) = −
18 Γ (cid:0) (cid:1) is achieved by using (9.3) with a = , b = and c = 2 to obtain(9.37) I = Γ (cid:0) (cid:1) ) (cid:2) ψ (cid:0) (cid:1) − ψ (cid:0) (cid:1)(cid:3) . Using (1.9) and (1.16) produces the stated result.
Example 9.16.
The usual application of (9.3) shows that . . is(9.38) Z ln x dx √ − x = 2 π √ (cid:0) ψ (cid:0) (cid:1) + γ (cid:1) , where we have used Γ( )Γ( ) = 2 π/ √
3. It remains to evaluate ψ ( ). The identity(1.16) gives(9.39) ψ (cid:0) (cid:1) − ψ (cid:0) (cid:1) = − π √ . To obtain a second relation among these quantities, we start from the identity(9.40) Γ(3 x ) = 3 x − / π Γ( x ) Γ( x + ) Γ( x + )that follows directly from (1.6), and differentiate logarithmically to obtain(9.41) ψ (3 x ) = ln 3 + 13 (cid:0) ψ ( x ) + ψ ( x + ) + ψ ( x + (cid:1) . The special case x = yields(9.42) ψ (cid:0) (cid:1) + ψ (cid:0) (cid:1) = − γ − . We conclude that(9.43) ψ ( ) = − γ −
32 ln 3 − π √ ψ ( ) = − γ −
32 ln 3 + π √ . This gives(9.45) Z ln x dx √ − x = − π √ (cid:18) ln 3 + π √ (cid:19) , as stated in . . . Example 9.17.
The evaluation of . . :(9.46) Z ln x dx √ − x = − π √ (cid:18) ln 3 − π √ (cid:19) , proceeds as in the previous example. The integral is identified as(9.47) I = Γ( )Γ( ) (cid:2) ψ (cid:0) (cid:1) + γ (cid:3) . The value (9.44) gives the rest.
Example 9.18.
The change of variables t = x yields(9.48) Z x p ln x dx √ − x = 116 Z t ( p − / (1 − t ) − / ln t dt. The last integral is evaluated using (9.3) with a = p +14 , b = and c = 1 to obtain(9.49) Z x p ln x dx √ − x = √ π
16 Γ( p +14 )Γ( p +34 ) (cid:20) ψ (cid:18) p + 14 (cid:19) − ψ (cid:18) p + 34 (cid:19)(cid:21) . The special case p = 4 n + 1 yields(9.50) Z x n +1 ln x dx √ − x = √ π n ! Γ( n + ) (cid:2) ψ ( n + ) − ψ ( n + 1) (cid:3) . The special case p = 4 n + 1 yields Z x n +1 ln x dx √ − x = √ π Γ( n + )16 n ! (cid:2) ψ ( n + ) − ψ ( n + 1) (cid:3) . Using (1.7), (1.11) and (1.14) yields . . in the form(9.51) Z x n +1 ln x dx √ − x = π (cid:0) nn (cid:1) n +3 − ln 2 + n X k =1 ( − k − k ! . The special case p = 4 n + 3 yields Z x n +3 ln x dx √ − x = √ π n !16Γ( n + ) (cid:2) ψ ( n + 1) − ψ ( n + ) (cid:3) . Using (1.7), (1.11) and (1.14) yields . . in the form(9.52) Z x n +3 ln x dx √ − x = 2 n − (2 n + 1) (cid:0) nn (cid:1) ln 2 + n +1 X k =1 ( − k k ! . Example 9.19.
The change of variables t = x n produces(9.53) Z ln x dx n √ − x n = 14 n Z t n − (1 − t ) − n ln t dt. Then (9.3) with a = n , b = 1 − n and c = 1 give the value(9.54) Z ln x dx n √ − x n = Γ( n ) Γ(1 − n )4 n Γ(1 − n ) (cid:2) ψ (cid:0) n (cid:1) − ψ (cid:0) − n (cid:1)(cid:3) . IGAMMA FUNCTION 19
Using (1.7) and (1.14) to obtain(9.55) Z ln x dx n √ − x n = − π B (cid:0) n , n (cid:1) n sin (cid:0) π n (cid:1) . This is . . in [2]. Example 9.20.
The change of variables t = x gives Z ln x dx n p x n − (1 − x ) = 14 Z t n − (1 − t ) − n ln t dt. Using (9.3) we obtain(9.56) Z ln x dx n p x n − (1 − x ) = Γ( n ) Γ(1 − n )Γ(1 − n ) (cid:2) ψ (cid:0) n (cid:1) − ψ (cid:0) − n (cid:1)(cid:3) . Proceeding as in the previous example, we obtain(9.57) Z ln x dx n p x n − (1 − x ) = − π B (cid:0) n , n (cid:1) sin (cid:0) π n (cid:1) . This is . . in [2].Some integrals in [2] have the form of the Corollary 9.2 after an elementarychange of variables. Example 9.21.
Formula . . in [2] states that(9.58) Z x a − ln(1 − x ) dx = − a ( ψ ( a + 1) + γ ) . This follows directly from (9.3) by the change of variables x − x . The same istrue for . . :(9.59) Z x a − (1 − x ) b − ln(1 − x ) dx = B ( a, b ) [ ψ ( b ) − ψ ( a + b )] . Example 9.22.
The change of variables t = e − x gives(9.60) Z ∞ xe − x (1 − e x ) n − dx = − Z (1 − t ) n − ln t dt. This latter integral is evaluated using (9.3) as(9.61) I = − √ π Γ( n + )4 n ! (cid:0) ψ (cid:0) (cid:1) − ψ ( n + 1) (cid:1) . Using (1.7) and (1.11) we obtain(9.62) Z ∞ xe − x (1 − e − x ) n − dx = (cid:0) nn (cid:1) π n +2 n X k =1 k ! . This appears as . . in [2]. 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 theseintegrals can be obtained using the incomplete beta function , defined by(10.1) β ( x ) := Z t x − dt x as it appears in . . . This function is related to the digamma function by theidentity(10.2) β ( x ) = 12 (cid:20) ψ (cid:18) x + 12 (cid:19) − ψ (cid:16) x (cid:17)(cid:21) . 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 ∞ (cid:18) e − x a −
11 + x b (cid:19) dxx = − γa , independently of b . Proof.
Write the integral as(11.2) Z ∞ (cid:16) e − x a − e − x b (cid:17) dxx + Z ∞ (cid:18) e − x b −
11 + x b (cid:19) dxx . 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 ∞ (cid:18) e − t −
11 + t (cid:19) dtt = − γb , according to (2.6). The formula has been established. (cid:3) Example 11.2.
The case a = 2 n and b = 2 n +1 gives . . :(11.4) Z ∞ (cid:18) exp( − x n ) −
11 + x n +1 (cid:19) dxx = − γ n . Example 11.3.
The case a = 2 n and b = 2 gives . . :(11.5) Z ∞ (cid:18) exp( − x n ) −
11 + x (cid:19) dxx = − γ n . Example 11.4.
The case a = 2 and b = 2 gives . :(11.6) Z ∞ (cid:18) e − x −
11 + x (cid:19) dxx = − γ . Example 11.5.
Finally, the change of variables t = ax yields(11.7) Z ∞ (cid:18) e − px −
11 + a x (cid:19) dxx = Z ∞ e − pt/a − e − t t dt + Z ∞ (cid:18) e − t −
11 + t (cid:19) dtt . IGAMMA FUNCTION 21
The first integral is ln ap according to (2.5), the second one is − γ . This gives theevaluation of . . :(11.8) Z ∞ (cid:18) e − px −
11 + a x (cid:19) dxx = γ + ln ap . 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 , 15,2007.[4] V. Moll. The integrals in Gradshteyn and Ryzhik. Part 6: The beta function.
Scientia , toappear.[5] V. Moll. The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function.
Scientia , in preparation.
Department of Mathematics, Tulane University, New Orleans, LA 70118
E-mail address : [email protected] Department of Mathematics, Tulane University, New Orleans, LA 70118
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