An Improper Integral with a Square Root

AN IMPROPER INTEGRAL WITH A SQUARE ROOT

FENG QI

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region

028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City 300387, China

Abstract. In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.

1. _Introduction

Letabe a positive number. Forn≥0, define

In = Z a

−a

xn

r

a+x

a−x dx. (1)

In [1, Section 3], Dana-Picard and Zeitoun computed I0 =aπ and found a closed

form ofIn forn∈Nin three steps:

(1) establishing a formula of recurrence between In andIn+1in terms of

Sn= Z π/2

−π/2

sinnθdθ; (2)

(2) establishing an equation forIn in terms ofSn;

(3) and establishing different expressions for odd values and even values ofn. The aim of this note is to discuss once again the sequenceIn and correct some errors and typos appeared in [1, Section 3].

2. _{Explicit and unified expressions for} In The sequenceIn can be computed by several methods below.

E-mail address:[email protected], [email protected].

2010 Mathematics Subject Classification. Primary 26A39; Secondary 11B65, 11B75, 11B83, 26A42, 33B15.

Key words and phrases. improper integral; explicit expression; unified expression; beta func-tion; Wallis ratio; integral representafunc-tion; Catalan number.

This paper was typeset usingAMS-LA_TEX.

Theorem 2.1. Forn≥0, the sequence In can be computed by

In =an+1π "

1 + (−1)n

n

1

B 1₂,n₂+

1 + (−1)n+1 n+ 1

1

B 1₂,n+1₂ #

, (3)

where

B(p, q) = Z 1

0

tp−1(1−t)q−1dt= Z ∞

0

tp−1

(1 +t)p+q dt=

Γ(p)Γ(q)

Γ(p+q) (4) and

Γ(z) = Z ∞

0

tz−1e−tdt

for<(p),<(q)>0 and<(z)>0 denote the Euler integrals of the second kinds (or say, the classical beta and gamma functions)respectively.

Proof. By some properties of definite integral and straightforward computation, we can write

In= Z 0

−a

xn

r

a+x a−x dx+

Z a

0 xn

r

a+x a−x dx

= Z 0

a (−y)n

s

a+ (−y)

a−(−y) d(−y) + Z a

0 xn

r

a+x a−x dx

= Z a

0

(−1)n_yn r_a

−y a+y dy+

Z a

0 xn

r

a+x a−x dx

= Z a

0 xn

(−1)n

r

a−x a+x +

r

a+x a−x

dx

= Z a

0

xn(a+x) + (−1)

n_(a−x) √

a2_−x2 dx

= Z a

0

xna[1 + (−1)

n_{] +x[1−(−1)}n_] √

a2_−x2 dx

=a[1 + (−1)n_] Z a

0

xn

√

a2_−x2 dx+ [1−(−1)

n_] Z a

0

xn+1

√

a2_−x2 dx.

In [8, Theorem 3.1], it was obtained that

Z a

0

xn √

a2_−x2 dx=

√

π anΓ

n

2 + 1 2

nΓ n₂ fora >0 andn≥0. Accordingly, considering

Γ ₁

2

=√π , (5)

we acquire

In =a[1 + (−1)n] √

π anΓ

n

2 + 1 2

nΓ n

2

+ [1−(−1)

n_]√_{π a}n+1 Γ

n+1 2 +

1 2

(n+ 1)Γ n+1 2

=√π an+1 [1 + (−1)n_]Γ n+1

2

nΓ n₂ + [1−(−1) n_] Γ

n

2+ 1

=an+1π

"

1 + (−1)n

n

Γ n+1₂

Γ 1₂

Γ n₂+

1−(−1)n

n+ 1

Γ n₂ + 1

Γ 1₂ Γ n+1₂

#

=an+1π

"

1 + (−1)n

n

1

B 1₂,n₂+

1 + (−1)n+1 n+ 1

1

B 1₂,n+1₂

#

.

The proof of Theorem 2.1 is complete.

Theorem 2.2. Forn≥0, the sequence In can be computed by

In= 1 2a

n+1

[1 + (−1)n_]B

1 2,

n+ 1 2

+1 + (−1)n+1

B

1 2,

n+ 2 2

. (6)

Proof. Changing the variable of integration byx=at in (1) and using some other properties of definite integral give

In= Z 1

−1

(at)n r

a+at a−atadt

=an+1

Z 1

−1 tn

r 1 +t

1−t dt

=an+1

Z 0

−1 tn

r 1 +t

1−t dt+

Z 1

0 tn

r 1 +t

1−t dt

=an+1

Z 1

0

(−s)n r

1−s

1 +s ds+

Z 1

0 tn

r 1 +t

1−t dt

=an+1

Z 1 0 tn (−1)n r 1−t

1 +t +

r 1 +t

1−t

dt

=an+1

Z 1

0 tn

(−1)n√1−t 1−t2 +

1 +t

√ 1−t2

dt

=an+1

[1 + (−1)n_] Z 1

0 tn √

1−t2 dt+ [1−(−1)

n_] Z 1

0

tn+1

√

1−t2 dt

=an+1

[1 + (−1)n_] Z π/2

0

sinns

p

1−sin2s

cossds

+ [1−(−1)n_] Z π/2

0

sinn+1s

p

1−sin2s

cossds

=an+1

[1 + (−1)n] Z π/2

0

sinnsds+ [1−(−1)n] Z π/2

0

sinn+1sds

.

Further making use of the formula

Z π/2

0

sintxdx= 1 2B

_{t+ 1} 2 ,

1 2

, t >−1

in [8, Remark 6.4] yields

In =an+1

[1 + (−1)n_]1 2B

₁

2,

n+ 1 2

+ [1−(−1)n_]1 2B

₁

2,

n+ 2 2 =1 2a n+1

[1 + (−1)n_{]B 1}

2,

n+ 1 2

+

1 + (−1)n+1

B

₁

2,

n+ 2 2

.

Corollary 2.1. Form≥0, the sequencesI2mandI2m+1can be explicitly computed

by

I2m=πa2m+1

(2m−1)!! (2m)!! and

I2m+1=πa2(m+1)

(2m+ 1)!! (2m+ 2)!!. Proof. From the recurrence relation

Γ(x+ 1) =xΓ(x), x >0 (7)

and the identity (5), we obtain

Γ

m+1 2

= (2m−1)!!

2m Γ

₁

2

= (2m−1)!! 2m

√

π .

By this equality and the last expression in (4), we derive

B

1 2,

n

2

=Γ

1 2

Γ n₂

Γ n+1₂ = 

  

  

Γ 1₂Γ(m)

Γ m+1₂, n= 2m Γ 1₂Γ m+1₂

Γ(m+ 1) , n= 2m+ 1

= 

  

  

√

π(m−1)!

(2m−1)!! 2m

√

π , n= 2m

√

π(2m₂−m1)!! √

π

m! , n= 2m+ 1 =



  

  

2(2m−2)!!

(2m−1)!!, n= 2m;

π(2m−1)!!

(2m)!! , n= 2m+ 1. Substituting this into (6) reveals

I2m= 1 2a

2m+1

2B

₁

2, 2m+ 1

2

=a2m+1π(2m−1)!!

(2m)!! and

I2m+1 =

1 2a

2(m+1)

2B

₁

2, 2m+ 3

2

=a2(m+1)π(2m+ 1)!!

(2m+ 2)!!.

The proof of Corollary 2.1 is complete.

3. _{Integral representations for the Catalan numbers}

The Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in various counting problems in combinatorial mathematics. Thenth Catalan number can be expressed in terms of the central binomial coefficients 2_nn

by

Cn= 1

n+ 1 _2n

n

. (8)

Theorem 3.1. Forn≥0 and a >0, the Catalan numbersCn can be represented by

Cn= 1

π

4n

n+ 1 1

a2n+1

Z a

−a

x2n

r

a+x a−x dx

= 1

π

22n+1 n+ 1

1

a2n Z a

0

x2n √

a2_−x2 dx

= 1

π

22n+1

n+ 1 Z π/2

0

sin2nxdx

and

Cn = 1

π

22n+1

2n+ 1 1

a2n+2

Z a

−a

x2n+1

r

a+x a−x dx

= 1

π

22n+2

2n+ 1 1

a2n+2

Z a

0

x2n+2

√

a2_−x2 dx

= 1

π

22n+2 2n+ 1

Z π/2

0

sin2n+2xdx.

(10)

Proof. From the recurrence relation (7) and the identity (5), it is not difficult to show that the Catalan numbersCncan be expressed in terms of the gamma function Γ by

Cn =

4n_{Γ(n+ 1/2)} √

πΓ(n+ 2), n≥0. This implies that

Cn= 1

π

4n

n+ 1B ₁

2, n+ 1 2

. (11)

Takingn= 2pin (6) and utilizing (11) lead to

I2p=a2p+1B ₁

2, 2p+ 1

2

=a2p+1πp+ 1

4p Cn which is equivalent to

Cn= 4n

n+ 1 1

a2n+1_πI2n=

1

π

4n

n+ 1 1

a2n+1

Z a

−a

x2n

r

a+x a−x dx.

The first formula in (9) thus follows.

By similar argument to the deduction of (11), we can discover

Cn=

4n+1

(2n+ 1)(2n+ 2) 1

B 1₂, n+ 1, n≥0. Employing this identity and settingn= 2p+ 1 in (3) figures out

I2p+1=a2p+2

2π

2p+ 2 1

B 1₂, p+ 1 =a

2p+2 2π

2p+ 2

(2p+ 1)(2p+ 2) 4p+1 Cp

which can be rearranged as

Cp= 1

a2p+2

1

π

22p+1

2p+ 1I2p+1= 1

π

1

a2p+2

22p+1

2p+ 1 Z a

−a

x2p+1

r

a+x a−x dx.

The first formula in (10) is thus proved.

The rest integral representations follow from techniques used in the proofs of Theorems (2.1) and (2.2) and from changing variable of integration.

4. _Remarks

Finally, we state several remarks on our main results.

Remark 4.2. Since

B

₁

2,

t+ 1 2

B

₁

2,

t

2

=2π

t

fort >0, the formulas (3) and (6) can be transferred to each other. However, the formula (6) looks simpler.

Remark 4.3. Considering (8), we can rewritten the integral representations in (9) and (10) as

_2n

n

= 1

π

4n

a2n+1

Z a

−a

x2n

r

a+x a−x dx

= 1

π

22n+1 a2n

Z a

0

x2n √

a2_−x2 dx

= 1

π2 2n+1

Z π/2

0

sin2nxdx

and

_2n

n

= 1

π

22n+1(n+ 1) 2n+ 1

1

a2n+2

Z a

−a

x2n+1

r

a+x a−x dx

= 1

π

22n+2_{(n+ 1)}

2n+ 1 1

a2n+2

Z a

0

x2n+2

√

a2_−x2 dx

= 1

π

22n+2_{(n+ 1)}

2n+ 1

Z π/2

0

sin2n+2xdx.

forn≥0.

Remark 4.4. It is well known that the Wallis ratio is defined by

Wn=

(2n−1)!! (2n)!! =

(2n)! 22n_(n!)2 =

1 √

π

Γ n+ 1/2

Γ(n+ 1) , n∈N. As a result, we have

I2m=πa2m+1Wm and

I2m+1=πa2m+2Wm+1

form≥0.

The Wallis ratio has been studied and applied by many mathematicians. For more information, please refer to [2, 3, 7, 12, 14], for example, and plenty of litera-ture therein.

Remark 4.5. In [3], the formula

Z π/2

0

sintxdx= √

π

2

Γ t+1₂

Γ t+2 2

, t >−1 (12)

was stated. See also [7, p. 16, Eq. (2.18)]. In [6, p. 142, Eq. 5.12.2], the formula

Z π/2

0

sin2a−1θcos2b−1θdθ=1

2B(a, b), <(a),<(b)>0 (13) was listed. By (12) or (13), we can find that the quantitySn defined in (2) is

Sn= Z 0

−π/2

sinnxdx+ Z π/2

0

= Z π/2

0

(−1)n_sinn_xdx+Z π/2

0

sinnxdx

= 1 + (−1) n

2 B

_{n+ 1} 2 ,

1 2

.

Remark 4.6. In [11, Theorem 2.3], among other things, the integral formulas

Z b

a _b−t

t−a

λ

dt= (b−a) λπ sin(λπ), Z b

a

b−t t−a

λ 1

t dt= π

sin(λπ)

b a

λ −1

,

Z b

a _b−t

t−a

λ 1

tk+1 dt= π

sin(λπ) _b

a

λ 1

ak k X

`=0

hλi`

`!

_k−1

`−1

1−a

b

`

forb > a >0,k∈N, andλ∈(−1,1)\ {0}were derived, where

hxin= 



 n−1

Q

k=0

(x−k), n≥1

1, n= 0

is called the falling factorial. In [11, Remark 4.4], the integral formula

Z b

a _b−t

t−a

λ₁

t ln b−t t−adt

= 

  

  

π

sin(λπ)

_b

a

λ ln b

a−πcot(λπ)

_b

a

λ −1

, λ6= 0 1

2

ln b

a

2

, λ= 0

was concluded from [11, Theorem 2.3]. By comparing the forms of these integrals andIn, we naturally propose a problem: can one explicitly compute the integrals

Z b

a _b

−t t−a

λ

tαdt and Z b

a

tα

_b −t t−a

λ

lnb−t

t−adt

for

α∈ (

R, b > a >0 N, b >0> a

andλ∈(−1,1)?

Remark 4.7. In recent years, the Catalan numbers Cn has been analytically gen-eralized and studied in [4, 5, 9, 10, 13, 15, 16, 17] and closely-related references therein.

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