# Special Functions

Authored by: Yu. A. Brychkov , O. I. Marichev , N. V. Savischenko

# Handbook of Mellin Transforms

Print publication date:  October  2018
Online publication date:  October  2018

Print ISBN: 9781138353350
eBook ISBN: 9780429434259

10.1201/9780429434259-3

#### Abstract

More formulas can be obtained from the corresponding sections due to the relations Γ φ x \begin{aligned} \Gamma \left(z\right)=\lim _{w\rightarrow \infty }\frac{w^z}{z}{\,}_1F_1\left(z;{\,}z+1;\,-w\right),\\ \Gamma \left(1-z\right)\Gamma \left(1+z\right)=\frac{z\pi }{\sin \left(z\pi \right)},\quad \Gamma \biggl (z+\frac{1}{2}\biggr )\Gamma \biggl (\frac{1}{2}-z\biggr )=\frac{\pi }{\cos \left(z\pi \right)},\\ \psi \left(z\right)=\left(z-1\right){\,}_3F_2\left(1,{\,}1,{\,}2-z;{\,}2,{\,}2;{\,}1\right)-\mathbf C ,\quad \psi \left(-z\right)=\frac{1}{z}+\pi \cot \left(z\pi \right)+\psi \left(z\right),\\ \psi ^{(n)}\left(z\right)=\left(-1\right)^{n+1} n!{\,} z^{-n-1} {\,}_{n+2}F_{n+1}\left(1,{\,}z,{\,}z,\ldots ,z;{\,}z+1,{\,}z+1,\ldots ,z+1;{\,}1\right),\\ \psi ^{(n)}\left(z\pm m\right)=\psi ^{(n)}\left(z\right)\pm \left(-1\right)^{n}n!\sum _{k=(1\mp 1)/2}^{m-(1\pm 1)/2}\frac{1}{\left(z\pm k\right)^{n+1}},\\ \zeta \left(s\right)=\text{ Li}_s\left(1\right),\quad \text{ Re}s>1;\quad \zeta \left(s,{\,}a+n\right)=\zeta \left(s,{\,}a\right)-\sum _{k=0}^{n-1}\frac{1}{\bigl (\left(a+k\right)^2\bigr )^{s/2}},\\ \zeta \left(s,{\,}a-n\right)=\zeta \left(s,{\,}a\right)+\sum _{k=0}^{n-1}\frac{1}{\bigl (\left(a+k-n\right)^2\bigr )^{s/2}}. \end{aligned}

#### 3.1  The Gamma ψ z $\Gamma \left(z\right)$ , Psi ζ z $\psi \left(z\right)$ , and Zeta Γ z = lim w → ∞ w z z 1 F 1 z ; z + 1 ; - w , Γ 1 - z Γ 1 + z = z π sin z π , Γ ( z + 1 2 ) Γ ( 1 2 - z ) = π cos z π , ψ z = z - 1 3 F 2 1 , 1 , 2 - z ; 2 , 2 ; 1 - C , ψ - z = 1 z + π cot z π + ψ z , ψ ( n ) z = - 1 n + 1 n ! z - n - 1 n + 2 F n + 1 1 , z , z , … , z ; z + 1 , z + 1 , … , z + 1 ; 1 , ψ ( n ) z ± m = ψ ( n ) z ± - 1 n n ! ∑ k = ( 1 ∓ 1 ) / 2 m - ( 1 ± 1 ) / 2 1 z ± k n + 1 , ζ s = Li s 1 , Re s > 1 ; ζ s , a + n = ζ s , a - ∑ k = 0 n - 1 1 ( a + k 2 ) s / 2 , ζ s , a - n = ζ s , a + ∑ k = 0 n - 1 1 ( a + k - n 2 ) s / 2 . $\zeta \left(z\right)$ Functions

More formulas can be obtained from the corresponding sections due to the relations

$Γ φ x$

#### 3.1.1  f x $\Gamma \left(\varphi \left(x\right)\right)$

No.

$F s$

$a x Γ x + b$

1

$a 1 - b μ a , s - 1 , b - 1$

$a , Re b , Re s > 0$

$ln x Γ x Γ x + 1 2$

2

$sec s π / 2 2 π s 1 - 2 - s - 1 Γ s ζ s + 1$

$0 < Re s < 1$

$x c a x Γ x + b + 1$

3

$a - b Γ s + c μ a , s + c - 1 , b$

$Re s + c > 0$

$θ 1 - x Γ 1 - ln x$

4

$ν e - s$

$θ 1 - x Γ b - ln x + 1$

5

$e bs ν e - s , b$

$θ 1 - x - ln x c Γ b - ln x + 1$

6

$Γ c + 1 e bs μ e - s , c , b$

$ψ a x + b$

#### 3.1.2  f x $\psi \left(ax+b\right)$

No.

$F s$

$ψ x + 1 + C$

1

$- π sin s π ζ 1 - s$

$- 1 < Re s < 0$

$ψ x + a - ψ x + b$

2

$π sin s π ζ 1 - s , b - ζ 1 - s , a$

$a , b > 0 ; 0 < Re s < 1$

$ln x - ψ x + 1$

3

$π sin s π ζ 1 - s$

$0 < Re s < 1$

$ln x - ψ ( x + 1 2 )$

4

$2 1 - s - 1 sin s π ζ s$

$0 < Re s < 1$

$ln x + 1 - ψ x + 1$

5

$π sin s π [ ζ 1 - s + 1 s ]$

$0 < Re s < 1$

$ψ ( n ) a x + b$

#### 3.1.3  f x $\psi ^{(n)}\left(ax+b\right)$

No.

$F s$

$1 x - ψ ′ x + 1$

1

$π s - 1 sin s π ζ 2 - s$

$1 < Re s < 2$

$1 x + 1 - ψ ′ x + 1$

2

$π s - 1 sin s π ζ 2 - s + 1 s - 1$

$0 < Re s < 2$

$ψ ( n ) x + 1$

3

$- 1 n - 1 π sin s π 1 - s n ζ 1 - s + n$

$0 < Re s < n$

$ζ ν , a x + b$

#### 3.1.4  f x ${\zeta \left(\nu ,{\,}ax+b\right)}$

No.

$F s$

$ζ ν , a x + b$

1

$a - s B s , ν - s ζ ν - s , b$

$Re ν , Re b > 0 ; 0 < Re s < Re ν - 1$

$ζ ν , x - 1 x ν$

2

$B s , ν - s ζ ν - s$

$0 < Re s < Re ν - 1$

$Li n z$

#### 3.2  The Polylogarithm Li n z = z n + 1 F n 1 , 1 , ⋯ , 1 ; 2 , 2 , ⋯ , 2 ; z , Li n - z = - G n + 1 , n + 1 1 , n + 1 z | 1 , 1 , ⋯ , 1 1 , 0 , ⋯ , 0 . ${\text{ Li}}_{n}\left(z\right)$

More formulas can be obtained from the corresponding sections due to the relations

$Li n b x$

#### 3.2.1  f x ${\text{ Li}}_{n}\left(bx\right)$ and algebraic functions

No.

$F s$

$θ a - x Li 2 x a$

1

$a s s 2 π 2 s 6 - ψ s + 1 - C$

$a > 0 ; Re s > - 1$

$Li n - a x$

2

$- 1 n π csc s π a s s n$

$- 1 < Re s < 0 ; | arg a | < π$

$θ a - x Li n - b x$

3

$a s + 1 b s s + 1 n + 1 F n 1 , 1 , … , 1 , s + 1 2 , … , 2 , s + 2 ; - a b$

$- a s + 1 b s n + 1 F n 1 , 1 , … , 1 2 , … , 2 ; - a b$

$a > 0 ; Re s > - 1 ; | arg b | < π$

$a - x + α - 1 Li n - b x$

4

$- a s + α b B α , s + 1 n + 2 F n + 1 1 , 1 , … , 1 , s + 1 ; - a b 2 , … , 2 , s + α + 1$

$a , Re α > 0 ; Re s > - 1 ; | arg b | < π$

$x - a + α - 1 Li n - b x$

5

$- a s + α b B α , - s - α n + 2 F n + 1 1 , 1 , … , 1 , s + 1 ; - a b 2 , … , 2 , s + α + 1$

$+ - 1 n + 1 π csc s + α π b s + α - 1 s + α - 1 n$

$× n + 1 F n 1 - α , 1 - s - α , … , 1 - s - α 2 - s - α , … , 2 - s - α ; - a b$

$a , Re α > 0 ; Re s + α < 1 ; | arg b | < π$

$1 x + a ρ Li n - b x$

6

$- a s - ρ + 1 b B s + 1 , ρ - s - 1$

$× n + 2 F n + 1 1 , 1 , … , 1 , s + 1 ; a b 2 , … , 2 , s - ρ + 2 + π b ρ - s ρ - s n$

$× csc s - ρ π n + 1 F n ρ , ρ - s , … , ρ - s ; a b ρ - s + 1 , … , ρ - s + 1$

$- 1 < Re s < Re ρ ; | arg a | , | arg b | < π$

$1 x - a Li n - b x$

7

$π a s b cot s π n + 1 F n 1 , 1 , … , 1 ; - a b 2 , 2 , … , 2$

$- π b 1 - s 1 - s n csc s π n + 1 F n 1 , 1 - s , … , 1 - s ; - a b 2 - s , … , 2 - s$

$a > 0 ; | Re s | < 1 ; | arg b | < π$

$a - x + α - 1 Li 2 - b x 2$

8

$- a s + α + 1 b B α , s + 2 5 F 4 1 , 1 , 1 , s + 2 2 , s + 3 2 ; - a 2 b 2 , 2 , s + α + 2 2 , s + α + 3 2$

$a , Re α > 0 ; Re s > - 2$

$1 x + a ρ Li 2 b x + a$

9

$a s - ρ - 1 b B s , 1 - s + ρ 4 F 3 1 , 1 , 1 , 1 - s + ρ 2 , 2 , ρ + 1 ; b a$

$0 < Re s < Re ρ + 1 ; | arg a | < π$

$1 x + a ρ Li 2 bx x + a$

10

$a s - ρ b B s + 1 , ρ - s 4 F 3 1 , 1 , 1 , s + 1 2 , 2 , ρ + 1 ; b$

$- 1 < Re s < Re ρ ; | arg a | < π$

$a - x + α - 1 Li 2 b x a - x$

11

$a s + α + 1 b B ( s + 1 , α + 1 ) 5 F 4 1 , 1 , 1 , s + 1 , α + 1 ; a 2 b 4 2 , 2 , s + α + 2 2 , s + α + 3 2$

$a > 0 ; Re s , Re α > - 1 ; | arg 4 - a 2 b | < π$

$Li n b x$

#### 3.2.2  f x $\text{ Li}_{n}\left(bx\right)$ and the logarithmic or inverse trigonometric functions

No.

$F s$

$θ a - x ln a + a - x x$

1

$π a s + 1 b 2 s s + 1 Γ [ s 2 s + 3 2 ] [ 3 F 2 1 , s + 1 , s + 1 2 s + 3 2 , s + 2 ; a b$

$× Li 2 b x$

$- s + 1 3 F 2 1 , 1 , s + 1 2 , 2 s + 3 2 ; a b + s s + 1 4 F 3 1 , 1 , 1 , s + 1 2 , 2 , 2 s + 3 2 ; a b ]$

$a > 0 ; Re s > - 2 ; | arg 1 - a b | < π$

$θ a - x arccos x a$

2

$π a s + 1 b 2 s 2 s + 1 Γ [ 2 s + 3 2 s + 2 ] [ 3 F 2 1 , s + 1 , 2 s + 3 2 s + 2 , s + 2 ; a b$

$× Li 2 b x$

$- s + 1 3 F 2 1 , 1 , s + 1 2 , 2 s + 3 2 ; a b + s s + 1 4 F 3 1 , 1 , 1 , 2 s + 3 2 2 , 2 , s + 2 ; a b ]$

$a > 0 ; Re s > - 1 ; | arg 1 - a b | < π$

$Ei z$

#### 3.3  The Exponential Integral Ei z = - e z Ψ 1 ; 1 ; - z + 1 2 ( ln z - ln 1 z ) - ln - z , Ei z = z 2 F 2 1 , 1 ; 2 , 2 ; z + 1 2 ( ln z - ln 1 z ) + C , Ei - z = - G 12 20 z | 1 0 , 0 , Ei - z = - e - z G 12 21 z | 0 0 , 0 . $\text{ Ei}\left(z\right)$

More formulas can be obtained from the corresponding sections due to the relations

$Ei φ x$

#### 3.3.1  f x $\text{ Ei}\left(\varphi \left(x\right)\right)$ and algebraic functions

No.

$F s$

$Ei - a x$

1

$- a - s s Γ s$

$a , Re s > 0$

$Ei - a x - b$

2

$- b a s Γ s Γ - s , b$

$a , Re s > 0 ; | arg b | < π$

$a - x + α - 1 Ei - b x$

3

$- a s + α b B s + 1 , α 3 F 3 s + 1 , 1 , 1 ; - a b s + α + 1 , 2 , 2$

$+ a s + α - 1 B s , α ψ s - ψ s + α + ln a b + C$

$a , Re α , Re s > 0 ; | arg b | < π$

$x - a + α - 1 Ei - b x$

4

$- a s + α b B α , - s - α 3 F 3 1 , 1 , s + 1 ; - a b 2 , 2 , s + α + 1$

$- b - s - α + 1 Γ s + α - 1 s + α - 1 2 F 2 1 - α , 1 - s - α ; - a b 2 - s - α , 2 - s - α$

$+ a s + α - 1 B α , 1 - s - α ψ 1 - s - ψ 1 - s - α + ln a b + C$

$a , Re b , Re α > 0 ; Re s + α < 1$

$1 x + a ρ Ei - b x$

5

$- a s - ρ + 1 b B s + 1 , ρ - s - 1 3 F 3 1 , 1 , s + 1 ; a b 2 , 2 , s - ρ + 2$

$+ b ρ - s Γ s - ρ ρ - s 2 F 2 ρ , ρ - s ; a b ρ - s + 1 , ρ - s + 1$

$+ a s - ρ B s , ρ - s ψ s - ψ ( ρ - s ) + ln a b + C$

$Re b > 0 ; 0 < Re s < ρ ; | arg a | < π$

$1 x + a Ei - b x$

6

$- b 1 - s Γ s - 1 s - 1 2 F 2 1 , 1 - s ; a b 2 - s , 2 - s$

$- π a s - 1 csc s π [ π cot s π + Γ 0 , - a b + ln 1 a + ln - a ]$

$Re b > 0 ; 0 < Re s < 1 ; | arg a | < π$

$1 x - a Ei - b x$

7

$π a s - 1 cot s π 2 π csc 2 s π - Ei - a b + b 1 - s 1 - s Γ s - 1$

$× 2 F 2 1 , 1 - s ; - a b 2 - s , 2 - s$

$a , Re b > 0 ; 0 < Re s < 1$

$a 2 - x 2 + α - 1 Ei - b x$

8

$a s + 2 α b 2 8 B α , s + 2 2 3 F 4 1 , 1 , s + 2 2 ; a 2 b 2 4 3 2 , 2 , 2 , s + 2 α + 2 2$

$- a s + 2 α - 1 b 2 B α , s + 1 2 2 F 3 1 2 , s + 1 2 ; a 2 b 2 4 3 2 , 3 2 , s + 2 α + 1 2$

$+ a s + 2 α - 2 2 B α , s 2 [ 1 2 ψ s 2 - 1 2 ψ s + 2 α 2$

$+ ln a b + C ]$

$a , Re α , Re s > 0 ; | arg b | < π$

$x 2 - a 2 + α - 1 Ei - b x$

9

$a s + 2 α b 2 8 B α , - s + 2 α 2 3 F 4 1 , 1 , s + 2 2 ; a 2 b 2 4 3 2 , 2 , 2 , s + 2 α + 2 2$

$- a s + 2 α - 1 b 2 B α , - s + 2 α - 1 2 2 F 3 1 2 , s + 1 2 ; a 2 b 2 4 3 2 , 3 2 , s + 2 α + 1 2$

$- Γ s + 2 α - 2 s + 2 α - 2 b - s - 2 α + 2 2 F 3 1 - α , - s + 2 α - 2 2 ; a 2 b 2 4 - s + 2 α - 3 2 , - s + 2 α - 4 2 , - s + 2 α - 4 2$

$+ a s + 2 α - 2 2 B α , - s + 2 α - 2 2 [ - 1 2 ψ - s + 2 α - 2 2 + ln a b$

$+ 1 2 ψ - s - 2 2 + C ]$

$a , Re b , Re α > 0 ; Re s + 2 α < 2$

$Ei φ x$

#### 3.3.2  f x $\text{ Ei}\left(\varphi \left(x\right)\right)$ and the exponential function

No.

$F s$

$e ± ax Ei ∓ ax$

1

$- π a s csc s π cot s π Γ s$

$a > 0 ; 0 < Re s < 1$

$e - a x Ei - b x$

2

$- Γ s s a + b s 2 F 1 1 , s ; a a + b s + 1$

$Re a + b , Re s > 0 ; | arg b | < π$

$e - a x Ei b x$

3

$- π a s cot s π Γ s + Γ s - 1 b a - b s - 1 2 F 1 1 , 1 ; b - a b 2 - s$

$Re a > b > 0 ; Re s > 0$

$e - a / x Ei - b x$

4

$a s Γ - s [ ab s + 1 2 F 3 1 , 1 ; a b 2 , 2 , s + 2 - ψ - s + ln a b + C ]$

$- b - s s Γ s 1 F 2 - s ; a b 1 - s , 1 - s$

$Re a , Re b > 0$

$e - a x Ei - b x$

5

$2 a 2 s + 1 b s + 1 / 2 Γ 2 s + 1 2 2 F 2 2 s + 1 2 , 2 s + 1 2 3 2 , 2 s + 3 2 ; a 2 4 b$

$- Γ s s b s 2 F 2 s , s ; a 2 4 b 1 2 , s + 1$

$( Re b , Re s > 0 ) or ( Re b = 0 ; Re a , Re s > 0 ) or ( Re b = Re a = 0 ; 0 < Re s < 2 ) ; ( Im b = 0 or ( Im b ≠ 0 ; Re a > 0 ) or ( Im b ≠ 0 ; Re a = 0 ; 2 Re s < 1 ) )$

$e ax Ei - a x - b$

6

$- π a - s sin s π Γ s , b$

$0 < Re s < 1$

$e ax [ Ei - 2 a x$

7

$a - s 2 Γ s ψ 2 - s 2 - ψ 1 - s 2$

$0 < Re s < 1 ; | arg a | < π$

$- Ei - a x ]$

$e bx Ei - u +$

8

$- π a ( s + 1 ) / 2 b 2 ( 1 - s ) / 2 cot s π 2 Γ s 2 K ( s + 1 ) / 2 a b$

$+ e - b x Ei u -$

$b , Re a > 0 ; 0 < Re s < 1$

$u ± = b ( x 2 + a 2 ± a )$

$Ei b x$

#### 3.3.3  δ = 1 0 $\text{ Ei}\left(bx\right)$ and hyperbolic or trigonometric functions

Notation: $f x$

.

No.

$F s$

$sin a x cos a x Ei - b x$

1

$- a δ s + δ b s + δ Γ s + δ 3 F 2 s + δ 2 , s + δ 2 , s + δ + 1 2 2 δ + 1 2 , s + δ + 2 2 ; - a 2 b 2$

$a , b > 0 ; Re s > - δ$

$sin a x cos a x Ei - b x$

2

$- 2 a δ 2 s + δ b s + δ / 2 Γ 2 s + δ 2 2 F 2 2 s + δ 2 , 2 s + δ 2 ; - a 2 4 b 2 δ + 1 2 , 2 s + δ + 2 2$

$Re a , Re a + b , Re s > 0$

$e bx sin a x Ei - b x$

3

$a 1 - s b Γ s - 1 cos s π 2 3 F 2 1 2 , 1 , 1 ; - a 2 b 2 2 - s 2 , 3 - s 2$

$- a 2 - s b 2 Γ s - 2 sin s π 2 3 F 2 1 , 1 , 3 2 ; - a 2 b 2 3 - s 2 , 4 - s 2$

$+ π csc s π a 2 + b 2 s / 2 Γ s sin s arctan a b$

$a > 0 ; - 1 < Re s < 2 ; | arg b | < π$

$e - b x sin a x Ei b x$

4

$π a 1 - s 2 b Γ 2 - s csc s π 2 3 F 2 1 2 , 1 , 1 ; - a 2 b 2 2 - s 2 , 3 - s 2$

$- π a 2 - s 2 b 2 Γ 3 - s sec s π 2 3 F 2 1 , 1 , 3 2 ; - a 2 b 2 3 - s 2 , 4 - s 2$

$- π cot s π a 2 + b 2 s / 2 Γ s sin s arctan a b$

$a , b > 0 ; - 1 < Re s < 2$

$e bx cos a x Ei - b x$

5

$- a 1 - s b Γ s - 1 sin s π 2 3 F 2 1 2 , 1 , 1 ; - a 2 b 2 2 - s 2 , 3 - s 2$

$- a 2 - s b 2 Γ s - 2 cos s π 2 3 F 2 1 , 1 , 3 2 ; - a 2 b 2 3 - s 2 , 4 - s 2$

$- π csc s π a 2 + b 2 s / 2 Γ s cos s arctan a b$

$a > 0 ; 0 < Re s < 2 ; | arg b | < π$

$e - b x cos a x Ei b x$

6

$- π a 1 - s 2 b Γ 2 - s sec s π 2 3 F 2 1 2 , 1 , 1 ; - a 2 b 2 2 - s 2 , 3 - s 2$

$- π a 2 - s 2 b 2 Γ 3 - s csc s π 2 3 F 2 1 , 1 , 3 2 ; - a 2 b 2 3 - s 2 , 4 - s 2$

$- π cot s π a 2 + b 2 s / 2 Γ s cos s arctan a b$

$a , b > 0 ; 0 < Re s < 2$

$sin a x sinh a x cos a x cosh a x$

7

$- a 2 δ b s + 2 δ s + 2 δ Γ s + 2 δ 5 F 4 s + 2 δ 4 , Δ 4 , s + 2 δ ; - 4 a 4 b 4 2 δ + 1 4 , 2 δ + 3 4 , 2 δ + 1 2 , s + 2 δ + 4 4$

$× Ei - b x$

$a , b > 0 ; Re s > - 2 δ$

$cos a x sinh a x sin a x cosh a x$

8

$± a 3 b - s - 3 3 s + 3 Γ s + 3 5 F 4 s + 3 4 , Δ 4 , s + 3 5 4 , 3 2 , 7 4 , s + 7 4 ; - 4 a 4 b 4$

$× Ei - b x$

$- a b - s - 1 s + 1 Γ s + 1 5 F 4 s + 1 4 , Δ 4 , s + 1 1 2 , 3 4 , 5 4 , s + 5 4 ; - 4 a 4 b 4$

$a , b > 0 ; Re s > - 1$

$e ax ln n x Ei b x$

#### 3.3.4  f x $e^{ax}{\,}\ln ^{n}x\text{ Ei}\left(bx\right)$

No.

$F s$

$ln a x Ei - b x$

1

$b - s s Γ s ln b a - ψ s + 1 s$

$Re a , Re b , Re s > 0$

$ln n x Ei - a x$

2

$- d n d s n Γ s a s s$

$Re a , Re s > 0$

$e ax ln x Ei - a x$

3

$π Γ s a s sin s π π cot s π - ψ s + ln a$

$0 < Re s < 1 ; | arg a | < π$

$e - a x ln x Ei - b x$

4

$Γ s a + b s [ ln a + b - ψ s Φ a a + b , 1 , s + Φ a a + b , 2 , s ]$

$Re a + b , Re s > 0 ; | arg b | < π$

$e - a x ln n x Ei - b x$

5

$- d n d s n Γ s a + b s Φ a a + b , 1 , s$

$Re a + b , Re s > 0 ; | arg b | < π$

$e ± a x ln n x Ei ∓ a x$

6

$- π d n d s n Γ s a s csc s π cot s π$

$0 < Re s < 1 ; | arg a | < π a > 0$

$Ei a x$

#### 3.3.5  Products of f x $\text{ Ei}\left(ax\right)$

No.

$F s$

$Ei 2 - a x$

1

$a - s Γ s 2 s - 1 s Φ 1 2 , 1 , s$

$a , Re s > 0$

$Ei - a x Ei - b x$

2

$Γ s a s [ bs a s + 1 4 F 3 1 , 1 , s + 1 , s + 1 2 , 2 , s + 2 ; - b a$

$+ 1 s 1 s - ψ s - C + ln a b ]$

$a + b , Re s > 0$

$e ax Ei 2 - a x$

3

$Γ s 2 a s 4 π 2 cos s π sin 2 s π + ψ ′ 1 - s 2 - ψ ′ 2 - s 2$

$a , Re s > 0$

$e - a x Ei - b x Ei b x$

4

$π s b s cot s π 2 Γ s 3 F 2 s 2 , s 2 , s + 1 2 1 2 , s + 2 2 ; a 2 b 2 + π a s + 1 b s + 1 tan s π 2 Γ s + 1$

$× 2 F 1 s + 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; a 2 b 2 - a 2 - s Γ s - 2 b 2 4 F 3 1 , 1 , 1 , 3 2 ; a 2 b 2 2 , 3 - s 2 , 4 - s 2$

$b , Re a , Re s > 0$

$ln a x Ei 2 - b x$

5

$2 1 - s b - s s Γ s { 2 s ψ s - 1 s - ln 2 2 F 1 1 , 1 ; - 1 s + 1$

$+ ln a b Φ 1 2 , 1 , s - Φ 1 2 , 2 , s }$

$b , Re a , Re s > 0$

$si z$

#### 3.4  The Sine Si z $\text{ si}\left(z\right)$ , ci z $\text{ Si}\left(z\right)$ , and Cosine si z = Si z - π 2 ; ci z = 1 2 Ei - i z + Ei i z , Re z > 0 ; si z = - π 2 ( z 2 z + 1 ) + i 2 Ei - i z - Ei i z , Re z ≠ 0 ; Si z = z 1 F 2 1 2 ; 3 2 , 3 2 ; - z 2 4 , ci z = - z 2 4 2 F 3 1 , 1 ; 2 , 2 , 3 2 ; - z 2 4 + ln z + C , ci z = - π 2 G 13 20 z 2 4 | 1 0 , 0 , 1 / 2 - ln z 2 2 + ln z , Si z = π z 2 2 z [ π - G 13 20 z 2 4 | 1 0 , 1 / 2 , 0 ] , Si z = π z 2 2 z G 13 11 z 2 4 | 1 1 / 2 , 0 , 0 , Si z = π z 4 G 13 11 z 2 4 | 1 / 2 0 , - 1 / 2 , - 1 / 2 . $\text{ ci}\left(z\right)$ Integrals

More formulas can be obtained from the corresponding sections due to the relations

$si a x$

#### 3.4.1  Si a x $\text{ si}\left(ax\right)$ , ci a x $\text{ Si}\left(ax\right)$ , and f x $\text{ ci}\left(ax\right)$

No.

$F s$

$si a x$

1

$- Γ s a s s sin s π 2$

$a > 0 ; 0 < Re s < 2$

$ci a x$

2

$- Γ s a s s cos s π 2$

$a > 0 ; 0 < Re s < 2$

$Si a x$

3

$- Γ s a s s sin s π 2$

$a > 0 ; - 1 < Re s < 0$

$si b x$

#### 3.4.2  ci b x $\text{ si}\left(bx\right)$ , f x $\text{ ci}\left(bx\right)$ , and algebraic functions

No.

$F s$

$a - x + α - 1 si b x$

1

$a s + α b B α , s + 1 3 F 4 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 4 3 2 , 3 2 , s + α + 1 2 , s + α + 2 2 - π 2 a s + α - 1 B α , s$

$a , b , Re α , Re s > 0$

$a - x + α - 1 ci b x$

2

$- a s + α + 1 b 2 4 B α , s + 2 4 F 5 1 , 1 , s + 2 2 , s + 3 2 ; - a 2 b 2 4 3 2 , 2 , 2 , s + α + 2 2 , s + α + 3 2$

$+ a s + α - 1 B α , s [ ψ s - ψ s + α + log a b + C ]$

$a , b , Re α , Re s > 0$

$a 2 - x 2 + α - 1 si b x$

3

$a s + 2 α - 1 b 2 B α , s + 1 2 2 F 3 1 2 , s + 1 2 ; - a 2 b 2 4 3 2 , 3 2 , s + 2 α + 1 2 - π a s + 2 α - 2 4 B α , s 2$

$a , b , Re α , Re s > 0$

$a 2 - x 2 + α - 1 ci b x$

4

$- a s + 2 α b 2 8 B α , s + 2 2 3 F 4 1 , 1 , s + 2 2 ; - a 2 b 2 4 3 2 , 2 , 2 , s + 2 α + 2 2$

$+ a s + 2 α - 2 2 B α , s 2 [ 1 2 ψ s 2 - 1 2 ψ s + 2 α 2 + ln a b + C ]$

$a , b , Re α , Re s > 0$

$1 ( x 2 + a 2 ) ρ si b x$

5

$- a s - 2 ρ + 3 b 3 36 B s + 3 2 , 2 ρ - s - 3 2 3 F 4 1 , 3 2 , s + 3 2 ; a 2 b 2 4 2 , 5 2 , 5 2 , s - 2 ρ + 5 2$

$+ a s - 2 ρ + 1 b 2 B s + 1 2 , 2 ρ - s - 1 2$

$- π a s - 2 ρ 4 B s 2 , 2 ρ - s 2 + b 2 ρ - s 2 ρ - s Γ s - 2 ρ$

$× sin s - 2 ρ π 2 2 F 3 ρ , 2 ρ - s 2 ; a 2 b 2 4 2 ρ - s + 1 2 , 2 ρ - s + 2 2 , 2 ρ - s + 2 2$

$b , Re a > 0 ; 0 < Re s < 2 Re ρ + 2$

$1 x 2 + a 2 ρ ci b x$

6

$- a s - 2 ρ + 2 b 2 8 B s + 2 2 , 2 ρ - s - 2 2 3 F 4 1 , 1 , s + 2 2 ; a 2 b 2 4 3 2 , 2 , 2 , s - 2 ρ + 4 2$

$+ a s - 2 ρ 2 B s 2 , 2 ρ - s 2 [ 1 2 ψ s 2 - 1 2 ψ 2 ρ - s 2 + ln a b + C ]$

$+ b 2 ρ - s 2 ρ - s Γ s - 2 ρ cos s - 2 ρ π 2 2 F 3 ρ , 2 ρ - s 2 ; a 2 b 2 4 2 ρ - s + 1 2 , 2 ρ - s + 2 2 , 2 ρ - s + 2 2$

$b , Re a > 0 ; 0 < Re s < 2 Re ρ + 2$

$1 x 2 - a 2 si b x$

7

$- π b 2 - s 2 2 - s Γ 3 - s sec s π 2 2 F 3 1 , 2 - s 2 ; - a 2 b 2 4 3 - s 2 , 4 - s 2 , 4 - s 2$

$+ π a s - 2 2 tan s π 2 Si a b + π 2 a s - 2 4 cot s π 2$

$a , b > 0 ; 0 < Re s < 4$

$1 x 2 - a 2 ci b x$

8

$- π b 2 - s 2 2 - s Γ 3 - s csc s π 2 2 F 3 1 , 2 - s 2 ; - a 2 b 2 4 3 - s 2 , 4 - s 2 , 4 - s 2$

$- π a s - 2 2 cot s π 2 ci a b + π 2 a s - 2 4 csc 2 s π 2$

$a , b > 0 ; 0 < Re s < 4$

$1 x + a ρ Si b x + a$

9

$a s - ρ - 1 b B s , 1 - s + ρ 3 F 4 1 2 , 1 - s + ρ 2 , 2 - s + ρ 2 ; - b 2 4 a 2 3 2 , 3 2 , ρ + 1 2 , ρ + 2 2$

$0 < Re s < Re ρ + 1 ; | arg a | < π$

$1 x + a ρ Si bx x + a$

10

$a s - ρ b B s + 1 , ρ - s 3 F 4 1 2 , s + 1 2 , s + 2 2 ; - b 2 4 3 2 , 3 2 , ρ + 1 2 , ρ + 2 2$

$- 1 < Re s < Re ρ ; | arg a | < π$

$1 x 2 + a 2 ρ$

11

$a s - 2 ρ - 1 b 2 B s + 1 2 , 1 - s + 2 ρ 2 3 F 4 1 2 , s + 1 2 , 1 - s + 2 ρ 2 ; - b 2 16 a 2 3 2 , 3 2 , ρ + 1 2 , ρ + 2 2$

$× Si bx x 2 + a 2$

$Re a > 0 ; - 1 < Re s < 2 Re ρ + 1$

$si b x$

#### 3.4.3  ci b x $\text{ si}\left(bx\right)$ , f x $\text{ ci}\left(bx\right)$ , and the exponential function

No.

$F s$

$e - a x si b x ci b x$

1

$± a Γ s + 1 b s + 1 s + 1 cos s π / 2 sin s π / 2 3 F 2 s + 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - a 2 b 2$

$- Γ s b s s sin s π / 2 cos s π / 2 3 F 2 s 2 , s 2 , s + 1 2 1 2 , s + 2 2 ; - a 2 b 2$

$b , Re a , Re s > 0$

$e - a x 2 si b x$

2

$- b 3 36 a ( s + 3 ) / 2 Γ s + 3 2 3 F 3 1 , 3 2 , s + 3 2 2 , 5 2 , 5 2 ; - b 2 4 a$

$+ b 2 a ( s + 1 ) / 2 Γ s + 1 2 - π 4 a s / 2 Γ s 2$

$b , Re a , Re s > 0$

$e - a x 2 ci b x$

3

$- b 2 8 a ( s + 2 ) / 2 Γ s + 2 2 3 F 3 1 , 1 , s + 2 2 3 2 , 2 , 2 ; - b 2 4 a$

$+ Γ s / 2 4 a s / 2 ψ s 2 + ln b 2 a + 2 C$

$b , Re a , Re s > 0$

$si b x$

#### 3.4.4  ci b x $\text{ si}\left(bx\right)$ , f x $\text{ ci}\left(bx\right)$ , and trigonometric functions

No.

$F s$

$sin a x si b x$

1

$b Γ s + 1 a s + 1 cos s π 2 3 F 2 1 2 , s + 1 2 , s + 2 2 3 2 , 3 2 ; b 2 a 2 - π Γ s 2 a s sin s π 2$

$0 < b < a ; - 1 < Re s < 2$

$sin a x si b x$

2

$- a Γ s + 1 b s + 1 s + 1 cos s π 2 3 F 2 s + 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; a 2 b 2$

$0 < a < b ; - 1 < Re s < 2$

$sin a x ci b x$

3

$b 2 Γ s + 2 4 a s + 2 sin s π 2 4 F 3 1 , 1 , s + 2 2 , s + 3 2 3 2 , 2 , 2 ; b 2 a 2$

$+ Γ s a s sin s π 2 C + ψ s + π 2 cot s π 2 + ln b a$

$0 < b < a ; - 1 < Re s < 2$

$sin a x ci b x$

4

$a Γ s + 1 b s + 1 s + 1 sin s π 2 3 F 2 s + 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; a 2 b 2$

$0 < a < b ; - 1 < Re s < 2$

$cos a x si b x$

5

$- b Γ s + 1 a s + 1 sin s π 2 3 F 2 1 2 , s + 1 2 , s + 2 2 3 2 , 3 2 ; b 2 a 2 - π Γ s 2 a s cos s π 2$

$0 < b < a ; 0 < Re s < 2$

$cos a x si b x$

6

$- Γ s b s s sin s π 2 3 F 2 s 2 , s 2 , s + 1 2 1 2 , s + 2 2 ; a 2 b 2$

$0 < a < b ; 0 < Re s < 2$

$cos a x ci b x$

7

$b 2 Γ s + 2 4 a s + 2 cos s π 2 4 F 3 1 , 1 , s + 2 2 , s + 3 2 3 2 , 2 , 2 ; b 2 a 2$

$+ Γ s a s cos s π 2 C + ψ s - π 2 tan s π 2 + ln b a$

$0 < b < a ; 0 < Re s < 2$

$cos a x ci b x$

8

$- Γ s b s s cos s π 2 3 F 2 s 2 , s 2 , s + 1 2 1 2 , s + 2 2 ; a 2 b 2$

$0 < a < b ; 0 < Re s < 2$

$sin a x ci a x$

9

$π 2 a s Γ s sec s π 2$

$a > 0 ; 0 < Re s < 1$

$- cos a x si a x$

$cos a x ci a x$

10

$- π 2 a s Γ s csc s π 2$

$a > 0 ; 0 < Re s < 2$

$+ sin a x si a x$

$cos a x ci a x$

11

$- π 2 a s cos s π 2 cot s π 2 Γ s$

$a > 0 ; 0 < Re s < 1$

$+ sin a x Si a x$

$sin a x ci a x$

12

$π 2 a s sin s π 2 tan s π 2 Γ s$

$a > 0 ; - 1 < Re s < 1$

$- cos a x Si a x$

$sin b x 2 + a 2$

13

$- π a ( s + 1 ) / 2 2 ( s + 3 ) / 2 b ( s - 1 ) / 2 csc s π 2 Γ s Γ 1 - s 2 J - ( s + 1 ) / 2 a b$

$× si b x 2 + a 2$

$- 2 ( s - 5 ) / 2 π 3 / 2 a ( s + 1 ) / 2 b ( s - 1 ) / 2 Γ s 2 [ sec s π 2 J ( s + 1 ) / 2 a b$

$+ cos b x 2 + a 2$

$+ csc s π 2 H ( s + 1 ) / 2 a b ] + π a s 2 s csc s π 2$

$× ci b x 2 + a 2$

$a , b > 0 ; 0 < Re s < 2$

$e - a x sin b x ci b x$

14

$π Γ s 2 b s sec s π 2 2 F 1 s 2 , s + 1 2 1 2 ; - a 2 b 2 + π a Γ s + 1 2 b s + 1 csc s π 2$

$- cos b x si b x$

$× 2 F 1 s + 1 2 , s + 2 2 3 2 ; - a 2 b 2 + Γ s - 1 a s - 1 b 3 F 2 1 2 , 1 , 1 2 - s 2 , 3 - s 2 ; - a 2 b 2$

$b , Re a , Re s > 0$

$e - a x cos b x ci b x$

15

$π a Γ s + 1 2 b s + 1 sec s π 2 2 F 1 s + 1 2 , s + 2 2 3 2 ; - a 2 b 2 - π Γ s 2 b s csc s π 2$

$+ sin b x si b x$

$× 2 F 1 s 2 , s + 1 2 1 2 ; - a 2 b 2 - Γ s - 2 a s - 2 b 2 3 F 2 3 2 , 1 , 1 3 - s 2 , 4 - s 2 ; - a 2 b 2$

$b , Re a , Re s > 0$

$Si b x$

#### 3.4.5  f x $\text{ Si}\left(bx\right)$ and the logarithmic or inverse trigonometric functions

No.

$F s$

$θ a - x ln a - x + a x$

1

$π a s + 1 b 2 s Γ s + 1 2 s + 3 2 [ 3 F 4 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 4 3 2 , 3 2 , 2 s + 3 4 , 2 s + 5 4$

$× Si b x$

$- 1 s + 1 3 F 4 s + 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 4 3 2 , 2 s + 3 4 , 2 s + 5 4 , s + 3 2 ]$

$a > 0 ; Re s > - 1$

$θ a - x ln a + a 2 - x 2 x$

2

$π a s + 1 b 2 s s + 1 Γ [ s + 1 2 s + 2 2 ] [ s + 1 2 F 3 1 2 , s + 1 2 ; - a 2 b 2 4 3 2 , 3 2 , s + 2 2$

$× Si b x$

$- 2 F 3 s + 1 2 , s + 1 2 ; - a 2 b 2 4 3 2 , s + 2 2 , s + 3 2 ]$

$a > 0 ; Re s > - 1$

$θ a - x arccos x a Si b x$

3

$π a s + 1 b 2 s + 1 Γ [ s + 2 2 s + 3 2 ] 3 F 4 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 4 3 2 , 3 2 , s + 3 2 , s + 3 2$

$a > 0 ; Re s > - 1$

$Si b x$

#### 3.4.6  si b x $\text{ Si}\left(bx\right)$ , ci b x $\text{ si}\left(bx\right)$ , Ei - a x r $\text{ ci}\left(bx\right)$ , and f x $\text{ Ei}\left(-ax^r\right)$

No.

$F s$

$Ei - a x Si b x$

1

$- b Γ s a s + 1 [ 3 F 2 1 2 , s + 1 2 , s + 2 2 3 2 , 3 2 ; - b 2 a 2 - 1 s + 1 3 F 2 s + 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - b 2 a 2 ]$

$a , b > 0 ; Re s > - 1$

$Ei - a x si b x$

2

$b 3 Γ s + 3 18 a s + 3 s + 3 5 F 4 1 , 3 2 , s + 3 2 , s + 3 2 , s + 4 2 2 , 5 2 , 5 2 , s + 5 2 ; - b 2 a 2 - b Γ s + 1 a s + 1 s + 1 + π Γ s 2 a s s$

$b , Re a , Re s > 0$

$Ei - a x ci b x$

3

$b 2 Γ s + 2 4 a s + 2 s + 2 5 F 4 1 , 1 , s + 2 2 , s + 2 2 , s + 3 2 2 , 2 , 3 2 , s + 4 2 ; - b 2 a 2$

$- Γ s a s s ψ s - 1 s + ln b a + C$

$b , Re a , Re s > 0$

$Ei - a x 2 si b x$

4

$b 3 18 a ( s + 3 ) / 2 s + 3 Γ s + 3 2 4 F 4 1 , 3 2 , s + 3 2 , s + 3 2 2 , 5 2 , 5 2 , s + 5 2 ; - b 2 4 a$

$- b a ( s + 1 ) / 2 s + 1 Γ s + 1 2 + π 2 a s / 2 s Γ s 2$

$a , Re b , Re s > 0$

$Ei - a x 2 ci b x$

5

$b 2 4 a s / 2 + 1 s + 2 Γ s + 2 2 4 F 4 1 , 1 , s + 2 2 , s + 2 2 2 , 2 , 3 2 , s + 4 2 ; - b 2 4 a$

$- Γ s / 2 a s / 2 s 1 2 ψ s 2 - 1 s + ln b a + C$

$b , Re a , Re s > 0$

$si 2 b x + ci 2 b x$

#### 3.4.7  f x ${\text{ si}^2\left(bx\right)+\text{ ci}^2\left(bx\right)}$ and trigonometric functions

No.

$F s$

$si 2 a x + ci 2 a x$

1

$π Γ s a s s csc s π 2$

$a > 0 ; 0 < Re s < 2$

$sin a x [ si 2 b x$

2

$- a 2 - s Γ s - 2 b 2 sin s π 2 4 F 3 1 , 1 , 1 , 3 2 ; a 2 b 2 2 , 3 - s 2 , 4 - s 2$

$+ ci 2 b x ]$

$+ π a Γ s + 1 b s + 1 s + 1 sec s π 2 3 F 2 s + 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; a 2 b 2$

$a , b > 0 ; - 1 < Re s < 2$

$cos a x [ si 2 b x$

3

$- a 2 - s Γ s - 2 b 2 cos s π 2 4 F 3 1 , 1 , 1 , 3 2 ; a 2 b 2 2 , 3 - s 2 , 4 - s 2$

$+ ci 2 b x ]$

$+ π Γ s b s s csc s π 2 3 F 2 s 2 , s + 1 2 , s 2 1 2 , s + 2 2 ; a 2 b 2$

$a , b > 0 ; 0 < Re s < 2$

$si b x$

#### 3.4.8  Products of ci b x $\text{ si}\left(bx\right)$ and f x $\text{ ci}\left(bx\right)$

No.

$F s$

$si a x si b x$

1

$- a - s - 1 b s + 1 cos s π 2 Γ s + 1 4 F 3 1 2 , s + 1 2 , s + 1 2 , s + 2 2 3 2 , 3 2 , s + 3 2 ; b 2 a 2$

$+ π 2 a s s sin s π 2 Γ s$

$0 < b < a ; 0 < Re s < 2$

$si a x ci b x$

2

$- a - s - 2 b 2 4 s + 2 sin s π 2 Γ s + 2 5 F 4 1 , 1 , s + 2 2 , s + 2 2 , s + 3 2 3 2 , 2 , 2 , s + 4 2 ; b 2 a 2$

$- Γ s a s s sin s π 2 ψ s + π 2 cot s π 2 - 1 s + ln b a + C$

$0 < b < a ; 0 < Re s < 2$

$si a x ci b x$

3

$a 3 b - s - 3 18 s + 3 sin s π 2 Γ s + 3 5 F 4 1 , 3 2 , s + 1 2 , s + 1 2 , s + 4 2 2 , 5 2 , 5 2 , s + 5 2 ; a 2 b 2$

$+ a b s + 1 s + 1 sin s π 2 Γ s + 1 + π 2 b s s cos s π 2 Γ s$

$0 < a < b ; 0 < Re s < 2$

$ci a x ci b x$

4

$- a - s - 2 b 2 4 s + 2 cos s π 2 Γ s + 2 5 F 4 1 , 1 , s + 2 2 , s + 2 2 , s + 3 2 3 2 , 2 , 2 , s + 4 2 ; b 2 a 2$

$- Γ s a s s cos s π 2 ψ s - π 2 tan s π 2 - 1 s + ln b a + C$

$0 < b < a ; 0 < Re s < 2$

$[ sin x ci 2 x$

5

$2 - s - 4 s Γ s { π 2 s 3 - cos s π sec s π 2 + 4 π 1 + cos s π$

$- cos x Si 2 x ] 2$

$× csc s π 2 + 4 s cos s π 2 ψ ′ s + 1 2 - ψ ′ s 2 }$

$- 2 < Re s < 0$

$[ sin x ci 2 x$

6

$2 - s - 3 Γ s { π 2 2 cos s π + 3 csc s π 2$

$- cos x Si 2 x ]$

$+ sin s π 2 3 ψ ′ s + 1 2 - 4 ψ ′ s - ψ ′ s 2 }$

$× [ cos x ci 2 x$

$- 1 < Re s < 1$

$+ sin x Si 2 x ]$

$shi z$

#### 3.5  Hyperbolic Sine chi z $\text{ shi}\left(z\right)$ and Cosine shi z = - i Si i z , shi z = z 1 F 2 1 2 ; 3 2 , 3 2 ; z 2 4 , chi z = ci i z - π i 2 , chi z = z 2 4 2 F 3 1 , 1 ; 2 , 2 , 3 2 ; z 2 4 + ln z + C , chi z = - π 2 G 13 20 - z 2 4 | 1 0 , 0 , 1 / 2 + 1 2 [ ln z - ln - z ] . $\text{ chi}\left(z\right)$ Integrals

More formulas can be obtained from the corresponding sections due to the relations

$shi b x$

#### 3.5.1  chi b x $\text{ shi}\left(bx\right)$ , f x $\text{ chi}\left(bx\right)$ , and algebraic functions

No.

$F s$

$a - x + α - 1 shi b x$

1

$a s + α b B α , s + 1 3 F 4 1 2 , s + 1 2 , s + 2 2 ; a 2 b 2 4 3 2 , 3 2 , s + α + 1 2 , s + α + 2 2$

$a , Re α , Re s > 0$

$a - x + α - 1 chi b x$

2

$a s + α + 1 b 2 4 B α , s + 2 4 F 5 1 , 1 , s + 2 2 , s + 3 2 ; a 2 b 2 4 3 2 , 2 , 2 , s + α + 2 2 , s + α + 3 2$

$+ a s + α - 1 B α , s [ ψ s - ψ s + α + log a b + C ]$

$a , Re α , Re s > 0$

$a 2 - x 2 + α - 1 shi b x$

3

$a s + 2 α - 1 b 2 B α , s + 1 2 2 F 3 1 2 , s + 1 2 ; a 2 b 2 4 3 2 , 3 2 , s + 2 α + 1 2$

$a , Re α , Re s > 0$

$a 2 - x 2 + α - 1 chi b x$

4

$a s + 2 α b 2 8 B α , s + 2 2 3 F 4 1 , 1 , s + 2 2 ; a 2 b 2 4 3 2 , 2 , 2 , s + 2 α + 2 2$

$+ a s + 2 α - 2 2 B α , s 2 [ 1 2 ψ s 2 - 1 2 ψ s + 2 α 2 + ln a b + C ]$

$a , Re α , Re s > 0$

$1 x + a ρ shi b x + a$

5

$a s - ρ - 1 b B s , 1 - s + ρ 3 F 4 1 2 , 1 - s + ρ 2 , 2 - s + ρ 2 ; b 2 4 a 2 3 2 , 3 2 , ρ + 1 2 , ρ + 2 2$

$0 < Re s < Re ρ + 1 ; | arg a | < π$

$1 x + a ρ shi bx x + a$

6

$a s - ρ b B s + 1 , ρ - s 3 F 4 1 2 , s + 1 2 , s + 2 2 ; b 2 4 3 2 , 3 2 , ρ + 1 2 , ρ + 2 2$

$- 1 < Re s < Re ρ ; | arg a | < π$

$1 x 2 + a 2 ρ shi bx x 2 + a 2$

7

$a s - 2 ρ - 1 b 2 B s + 1 2 , 1 - s + 2 ρ 2 3 F 4 1 2 , s + 1 2 , 1 - s + 2 ρ 2 ; b 2 16 a 2 3 2 , 3 2 , ρ + 1 2 , ρ + 2 2$

$Re a > 0 ; - 1 < Re s < 2 Re ρ + 1$

$shi b x$

#### 3.5.2  chi b x $\text{ shi}\left(bx\right)$ , f x $\text{ chi}\left(bx\right)$ , and the exponential function

No.

$F s$

$e - a x shi b x$

1

$b 3 18 a s + 3 Γ s + 3 4 F 3 1 , 3 2 , s + 3 2 , s + 4 2 2 , 5 2 , 5 2 ; b 2 a 2 + b a s + 1 Γ s + 1$

$Re a > | Re b | ; Re s > 0$

$e - a x chi b x$

2

$b 2 4 a s + 2 Γ s + 2 4 F 3 1 , 1 , s + 2 2 , s + 3 2 3 2 , 2 , 2 ; b 2 a 2 + Γ s a s [ ψ s + ln b a + C ]$

$Re a > | Re b | ; Re s > 0$

$e - a x 2 shi b x$

3

$b 3 36 a ( s + 3 ) / 2 Γ s + 3 2 3 F 3 1 , 3 2 , s + 3 2 2 , 5 2 , 5 2 ; b 2 4 a + b 2 a ( s + 1 ) / 2 Γ s + 1 2$

$Re a , Re s > 0 ; | arg b | < π$

$e - a x 2 chi b x$

4

$b 2 8 a s / 2 + 1 Γ s + 2 2 3 F 3 1 , 1 , s + 2 2 3 2 , 2 , 2 ; b 2 4 a$

$+ 1 2 a s / 2 Γ ( s 2 ) [ 1 2 ψ s 2 + ln b a + C ]$

$Re a , Re s > 0 ; | arg b | < π$

$shi b x$

#### 3.5.3  f x $\text{ shi}\left(bx\right)$ and the logarithmic or inverse trigonometric functions

No.

$F s$

$θ a - x ln a - x + a x$

1

$π a s + 1 b 2 s Γ s + 1 2 s + 3 2 [ 3 F 4 1 2 , s + 1 2 , s + 2 2 ; a 2 b 2 4 3 2 , 3 2 , 2 s + 3 4 , 2 s + 5 4$

$× shi b x$

$- 1 s + 1 3 F 4 s + 1 2 , s + 1 2 , s + 2 2 ; a 2 b 2 4 3 2 , 2 s + 3 4 , 2 s + 5 4 , s + 3 2 ]$

$a > 0 ; Re s > - 1$

$θ a - x arccos x a shi b x$

2

$π a s + 1 b 2 s + 1 Γ [ s + 2 2 s + 3 2 ] 3 F 4 1 2 , s + 1 2 , s + 2 2 ; a 2 b 2 4 3 2 , 3 2 , s + 3 2 , s + 3 2$

$a > 0 ; Re s > - 1$

$erf z$

#### 3.6  erfc z ${\text{ erf}\left(z\right)}$ , erfi z ${\text{ erfc}\left(z\right)}$ , and erf z erfc z = 1 π γ 1 / 2 , z 2 Γ 1 / 2 , z 2 , erf z erfi z = 2 z π 1 F 1 1 2 ; 3 2 ; ∓ z 2 , erf z erfi z = z ± z 2 [ 1 - e - z 2 π Ψ 1 2 ; 1 2 ; ± z 2 ] , erf z = - i erfi i z = 1 - erfc z , erfc z = z z 2 [ e - z 2 π Ψ 1 2 ; 1 2 ; z 2 - 1 ] + 1 , erfc z = 1 - 2 z π 1 F 1 1 2 ; 3 2 ; - z 2 , erf z = 2 z - i z 2 C - i z 2 - i S - i z 2 , erf z = z π z 2 G 12 11 z 2 | 1 1 / 2 , 0 , erfc z = 1 π G 12 20 z | 1 0 , 1 / 2 , erfi z = z - π z 2 G 12 11 - z 2 | 1 1 / 2 , 0 . ${\text{ erfi}\left(z\right)}$

More formulas can be obtained from the corresponding sections due to the relations

$erf a x + b$

#### 3.6.1  erfc a x + b x - 1 ${\text{ erf}\left(ax+b\right)}$ , f x ${\text{ erfc}\left(ax+b{x}^{-1}\right)}$

No.

$F s$

$erf a x + b - erf c x + b$

1

$e - b 2 c - s - a - s 2 s π Γ s Ψ s + 1 2 1 2 ; b 2$

$Re s > 0 ; | arg a | , | arg c | < π / 4$

$erf a x + b - erf c x + d$

2

$Γ s 2 ( s - 1 ) / 2 π c - s e - d 2 / 2 D - s - 1 2 d - a - s e - b 2 / 2 D - s - 1 2 b$

$Re s > 0 ; | arg a | , | arg c | < π / 4$

$erfc a x ± b x$

3

$2 b π s b a ( s - 1 ) / 2 e ∓ 2 a b K ( s + 1 ) / 2 2 a b ∓ K ( s - 1 ) / 2 2 a b$

$b > 0 ; | arg a | < π / 4$

$erf b x$

#### 3.6.2  erfc b x ${\text{ erf}\left(bx\right)}$ , f x ${\text{ erfc}\left(bx\right)}$ , and algebraic functions

No.

$F s$

$erf a x erfc a x$

1

$∓ a - s π s Γ s + 1 2$

$- 1 < Re s < 0 Re s > 0 ; | arg a | < π / 4$

$a - x + α - 1 erf b x erfc b x$

2

$± 2 a s + α b π B s + 1 , α 3 F 3 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 3 2 , s + α + 2 2 , s + α + 1 2$

$+ 1 ∓ 1 2 a s + α - 1 B s , α$

$a , Re α > 0 ; Re s > - 1 ± 1 / 2$

$x - a + α - 1 erf b x erfc b x$

3

$± 2 a s + α b π B α , - s - α 3 F 3 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 3 2 , s + α + 2 2 , s + α + 1 2$

$± Γ s + α 2 π b s + α - 1 1 - s - α 3 F 3 1 - α 2 , 2 - α 2 , 1 - s - α 2 ; - a 2 b 2 1 2 , 2 - s - α 2 , 3 - s - α 2$

$± a 1 - α Γ s + α - 1 2 π b s + α - 2 2 - s - α 3 F 3 2 - α 2 , 3 - α 2 , 2 - s - α 2 ; - a 2 b 2 3 2 , 3 - s - α 2 , 4 - s - α 2$

$+ 1 ∓ 1 2 a s + α - 1 B α , 1 - α - s$

$Re α > 0 , a > 0 ; Re s + α < 1 Re a > 0 ; | arg b | < π / 4$

$a 2 - x 2 + α - 1 erf b x erfc b x$

4

$± a s + 2 α - 1 b π B s + 1 2 , α 2 F 2 1 2 , s + 1 2 ; - a 2 b 2 3 2 , s + 2 α + 1 2$

$+ 1 ∓ 1 4 a s + 2 α - 2 B s 2 , α$

$a , Re α > 0 ; Re s > - 1 ± 1 / 2$

$x 2 - a 2 + α - 1 erf b x erfc b x$

5

$± a s + 2 α - 1 b π B 1 - s - 2 α 2 , α 2 F 2 1 2 , s + 1 2 ; - a 2 b 2 3 2 , s + 2 α + 1 2$

$± b 2 - s - 2 α π 2 - s - 2 α Γ s + 2 α - 1 2$

$× 2 F 2 1 - α , 2 - s - 2 α 2 ; - a 2 b 2 3 - s - 2 α 2 , 4 - s - 2 α 2$

$+ 1 ∓ 1 4 a s + 2 α - 2 B 2 - s - 2 α 2 , α$

$Re α > 0 , a > 0 ; Re s + 2 α < 2 a > 0 ; | arg b | < π / 4$

$1 x + a ρ erf b x erfc b x$

6

$± 2 a s - ρ + 1 b π B s + 1 , ρ - s - 1 3 F 3 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 3 2 , s - ρ + 2 2 , s - ρ + 3 2$

$± 1 π b s - ρ ρ - s Γ s - ρ + 1 2 3 F 3 ρ 2 , ρ + 1 2 , ρ - s 2 ; - a 2 b 2 1 2 , ρ - s + 1 2 , ρ - s + 2 2$

$∓ ρ a π b s - ρ - 1 ρ - s + 1 Γ s - ρ 2$

$× 3 F 3 ρ + 1 2 , ρ + 2 2 , ρ - s + 1 2 ; - a 2 b 2 3 2 , ρ - s + 2 2 , ρ - s + 3 2 + 1 ∓ 1 2 a s - ρ B s , ρ - s$

$- 1 < Re s < Re ρ Re s > 0 ; | arg a | , 4 | arg b | < π$

$1 x - a erf b x erfc b x$

7

$∓ π a s - 1 b cot s π erf a b ± Γ s 2 π b s - 1 1 - s 2 F 2 1 , 1 - s 2 ; - a 2 b 2 2 - s 2 , 3 - s 2$

$± a Γ s - 1 2 π b s - 2 2 - s 2 F 2 1 , 2 - s 2 ; - a 2 b 2 3 - s 2 , 4 - s 2 - π ∓ π 2 a s - 1 cot s π$

$a > 0 ; | Re s | < 1 ; | arg b | < π / 4$

$1 x 2 + a 2 ρ erf b x erfc b x$

8

$± a s - 2 ρ + 1 b π B s + 1 2 , 2 ρ - s - 1 2 2 F 2 1 2 , s + 1 2 ; a 2 b 2 3 2 , s - 2 ρ + 3 2$

$± b 2 ρ - s π 2 ρ - s Γ s - 2 ρ + 1 2 2 F 2 ρ , 2 ρ - s 2 ; a 2 b 2 2 ρ - s + 1 2 , 2 ρ - s + 2 2$

$+ 1 ∓ 1 4 a s - 2 ρ B s 2 , 2 ρ - s 2$

$Re a > 0 ; - 1 < Re s < 2 Re ρ Re s > 0 ; | arg b | < π / 4$

$1 x 2 - a 2 erf b x erfc b x$

9

$± π a s - 2 2 tan s π 2 erf a b ± b 2 - s π 2 - s Γ s - 1 2$

$× 2 F 2 1 , 2 - s 2 ; - a 2 b 2 3 - s 2 , 4 - s 2 - 1 ∓ 1 π a s - 2 4 cot s π 2$

$a > 0 ; - 1 < Re s < 2 Re s > 0 ; | arg b | < π / 4$

$a x 2 + b n erfc c x$

10

$b n π c s s Γ s + 1 2 3 F 1 - n , s 2 , s + 1 2 s + 2 2 ; - a b c 2$

$Re s > 0 ; | arg c | < π / 4$

$erf b x$

#### 3.6.3  erfc b x ${\text{ erf}\left(bx\right)}$ , f x ${\text{ erfc}\left(bx\right)}$ , and the exponential function

No.

$F s$

$× Γ s + 2 2 2 F 2 s + 1 2 , s + 2 2 3 2 , s + 3 2 ; a 2 4 b 2 + 1 ± 1 2 a s Γ s$

$Re a > 0 , Re s > - 1 Re s > 0 ; | arg b | < π / 4$

$e - a x 2 erf b x erfc b x$

2

$± b π a ( s + 1 ) / 2 Γ s + 1 2 2 F 1 1 2 , s + 1 2 3 2 ; - b 2 a + 1 ∓ 1 4 a s / 2 Γ s 2$

$Re a > 0 ; Re s > - 1 ± 1 / 2 ; | arg b | < π / 4$

$e a 2 x 2 erfc a x$

3

$a - s 2 Γ s 2 sec s π 2$

$0 < Re s < 1 ; | arg a | < π / 4$

$e - a 2 x 2 erfi a x$

4

$π 2 a s Γ 2 - s 2 sec s π 2$

$| Re s | < 1 ; | arg a | < π / 4$

$e a x 2 erfc b x$

5

$b - s π s Γ s + 1 2 2 F 1 s 2 , s + 1 2 s + 2 2 ; a b 2$

$Re b 2 - a , Re s > 0 ; | Re s | < 1 ; | arg b | < π / 4$

$e - a / x erf b x erfc b x$

6

$1 ∓ 1 2 a s Γ - s ± 2 a s + 1 b π Γ - s - 1 1 F 3 1 2 ; - a 2 b 2 4 3 2 , s + 2 2 , s + 3 2$

$∓ 1 π b s s Γ s + 1 2 1 F 3 - s 2 ; - a 2 b 2 4 1 2 , 1 - s 2 , 2 - s 2$

$± a π b s - 1 s - 1 Γ s 2 1 F 3 1 - s 2 ; - a 2 b 2 4 3 2 , 2 - s 2 , 3 - s 2$

$Re a > 0 ; Re s < 0 Re a > 0 ; | arg b | < π / 4$

$e - a / x 2 erf b x erfc b x$

7

$1 ∓ 1 4 a s / 2 Γ - s 2 ± a ( s + 1 ) / 2 b π Γ - s + 1 2 1 F 2 1 2 ; a b 2 3 2 , s + 3 2$

$∓ 1 π b s s Γ s + 1 2 1 F 2 - s 2 ; a b 2 1 - s 2 , 2 - s 2$

$Re a > 0 ; Re s < 0 Re a > 0 ; | arg b | < π / 4$

$e - a x - b 2 x 2 erfi b x$

8

$Γ s - 1 π a s - 1 b 2 F 2 1 2 , 1 ; a 2 4 b 2 2 - s 2 , 3 - s 2 + Γ s / 2 2 b s tan s π 2 1 F 1 s 2 ; a 2 4 b 2 1 2$

$+ a 2 b s + 1 Γ s + 1 2 cot s π 2 1 F 1 s + 1 2 ; a 2 4 b 2 3 2$

$Re a > 0 ; Re s > - 1 ; | arg b | < π / 4$

$e - a x + b 2 x 2 erfc b x$

9

$Γ s - 1 π a s - 1 b 2 F 2 1 2 , 1 ; - a 2 4 b 2 2 - s 2 , 3 - s 2 + Γ s 2 2 b s sec s π 2 1 F 1 s 2 ; - a 2 4 b 2 1 2$

$+ a 2 b s + 1 Γ s + 1 2 csc s π 2 1 F 1 s + 1 2 ; - a 2 4 b 2 3 2$

$Re a , Re s > 0 ; | arg b | < π / 4$

$e - a x - b x 2 erf c x$

10

$c π b ( s + 1 ) / 2 Γ s + 1 2 Ψ 1 s + 1 2 , 1 2 , 3 2 , 1 2 ; - c 2 b ; a 2 4 b$

$- ac π b ( s + 2 ) / 2 Γ s + 2 2 Ψ 1 s + 2 2 , 1 2 , 3 2 , 3 2 ; - c 2 b ; a 2 4 b$

$Re b , Re b + c 2 > 0 ; Re s > - 1$

$e - b 2 x 2 - a / x 2 erfi b x$

11

$- π a s / 4 2 b s / 2 sec s π 2 L s / 2 2 b a - I - s / 2 2 b a$

$Re a > 0 ; Re s < 1 ; s ≠ - 1 , - 3 , … ; | arg b | < π / 4$

$e b 2 x 2 - a / x 2 erfc b x$

12

$π a s / 4 2 b s / 2 sec s π 2 H s / 2 2 b a - Y s / 2 2 b a$

$Re a > 0 ; Re s < 1 ; s ≠ - 1 , - 3 , … ; | arg b | < π / 4$

$e a 2 x 2 erfc a x + b$

13

$Γ s π 2 a s Γ 1 - s 2 , b 2$

$0 < Re s < 1 ; | arg a | < π / 4$

$e - a 2 x erfi a x$

14

$a - 2 s Γ 1 - 2 s 2 , 2 s + 1 2 1 - s$

$0 < | Re s | < 1 / 2 ; | arg a | < π / 4$

$θ a - x e bx erf c a - x$

15

$a s + 1 / 2 c Γ s 2 s + 3 2 Φ 2 s , 1 2 ; 2 s + 3 2 ; a b , - a c 2$

$a , Re s > 0$

$erf b x$

#### 3.6.4  erfc b x $\text{ erf}\left(bx\right)$ , erfi b x $\text{ erfc}\left(bx\right)$ , f x $\text{ erfi}\left(bx\right)$ , and algebraic or the exponential functions

No.

$F s$

$a - x + α - 1 e b 2 x 2$

1

$± 2 a s + α b π B s + 1 , α 3 F 3 1 , s + 1 2 , s + 2 2 ; a 2 b 2 3 2 , s + α + 1 2 , s + α + 2 2$

$× erf b x erfc b x$

$+ 1 ∓ 1 2 a s + α - 1 B s , α 2 F 2 s 2 , s + 1 2 ; a 2 b 2 s + α 2 , s + α + 1 2$

$a , Re α > 0 ; Re s > - 1 ± 1 / 2$

$a 2 - x 2 + α - 1 e b 2 x 2$

2

$± a s + 2 α - 1 b π B s + 1 2 , α 2 F 2 1 , s + 1 2 ; a 2 b 2 3 2 , s + 2 α + 1 2$

$× erf b x erfc b x$

$+ 1 ∓ 1 4 a s + 2 α - 2 B s 2 , α 1 F 1 s 2 ; a 2 b 2 s + 2 α 2 ;$

$a , Re α > 0 ; Re s > - 1 ± 1 / 2$

$x 2 - a 2 + α - 1 e ∓ b 2 x 2$

3

$± a s + 2 α - 1 b π B 1 - s - 2 α 2 , α 2 F 2 1 , s + 1 2 ; ∓ a 2 b 2 3 2 , s + 2 α + 1 2$

$× erfi b x erfc b x$

$+ 1 ∓ 1 4 a s + 2 α - 2 B 2 - s - 2 α 2 , α 1 F 1 s 2 ; a 2 b 2 s + 2 α 2 ;$

$± b 2 - s - 2 α 2 tan s + 2 α π / 2 sec s + 2 α π / 2$

$× Γ s + 2 α - 2 2 1 F 1 1 - α ; ∓ a 2 b 2 4 - s - 2 α 2$

$a , Re α > 0 ; Re s + 2 α < 3 ; | arg b | < π / 4$

$e ∓ b 2 x 2 x + a ρ erfi b x erfc b x$

4

$± 2 a s - ρ + 1 b π B s + 1 , ρ - s - 1 3 F 3 1 , s + 1 2 , s + 2 2 ; ∓ a 2 b 2 3 2 , s - ρ + 2 2 , s - ρ + 3 2$

$∓ b ρ - s 2 tan ρ - s π / 2 sec ρ - s π / 2 Γ s - ρ 2$

$× 2 F 2 ρ 2 , ρ + 1 2 ; ∓ a 2 b 2 1 2 , ρ - s + 2 2 ± ρ a b ρ - s + 1 2 cot s - ρ π / 2 csc s - ρ π / 2$

$× Γ s - ρ - 1 2 2 F 2 ρ + 1 2 , ρ + 2 2 ; ∓ a 2 b 2 3 2 , 3 - s + ρ 2$

$+ 1 ∓ 1 2 a s - ρ B s , ρ - s 2 F 2 s 2 , s + 1 2 ; a 2 b 2 s - ρ + 1 2 , s - ρ + 2 2$

$- 1 ± 1 / 2 < Re s < Re ρ + 1 ; | arg a | , 4 | arg b | < π$

$e - b 2 x 2 x + a erfi b x$

5

$a s - 1 2 e - a 2 b 2 i s - 1 cot s π 2 Γ s + 1 2 γ 1 - s 2 , - a 2 b 2$

$- i s tan s π 2 Γ s 2 γ 2 - s 2 , - a 2 b 2 - 2 π sin s π erfi a b$

$- 1 < Re s < 2 ; | arg a | , 4 | arg b | < π$

$e - b 2 x 2 x - a erfi b x$

6

$- π a s - 1 e - a 2 b 2 cot s π erfi a b$

$- b 1 - s 2 cot s π 2 Γ s - 1 2 1 F 1 1 ; - a 2 b 2 3 - s 2$

$+ a b 2 - s 2 tan s π 2 Γ s - 2 2 1 F 1 1 ; - a 2 b 2 4 - s 2$

$a > 0 ; - 1 < Re s < 2 ; | arg b | < π / 4$

$e ∓ b 2 x 2 x 2 + a 2 ρ erfi b x erfc b x$

7

$± a s - 2 ρ + 1 b π B s + 1 2 , 2 ρ - s - 1 2 2 F 2 1 , s + 1 2 ; ± a 2 b 2 3 2 , s - 2 ρ + 3 2$

$∓ b 2 ρ - s 2 tan 2 ρ - s π / 2 sec 2 ρ - s π / 2 Γ s - 2 ρ 2 1 F 1 ρ ; ± a 2 b 2 2 - s + 2 ρ 2$

$+ 1 ∓ 1 4 a s - 2 ρ B s 2 , 2 ρ - s 2 1 F 1 s 2 ; - a 2 b 2 s - 2 ρ + 2 2 ;$

$Re a > 0 ; - 1 ± 1 / 2 < Re s < 2 Re ρ + 1 ; | arg b | < π / 4$

$e b 2 x 2 x 2 + a 2 erfc b x$

8

$π a s - 2 2 e - a 2 b 2 sec s π 2 [ cot s π 2 - erfi a b$

$+ i s - 2 π Γ s 2 γ 2 - s 2 , - a 2 b 2 ]$

$Re a > 0 ; 0 < Re s < 3 ; | arg b | < π / 4$

$e - b 2 x 2 x 2 + a 2 erfi b x$

9

$π a s - 2 2 e a 2 b 2 sec s π 2 [ erf a b - 1 Γ 2 - s 2 γ 2 - s 2 , a 2 b 2 ]$

$Re a > 0 ; - 1 < Re s < 3 ; | arg b | < π / 4$

$e b 2 x 2 x 2 - a 2 erfc b x$

10

$π a s - 2 2 e a 2 b 2 tan s π 2 erfc a b - π a s - 2 sin s π e a 2 b 2$

$- b 2 - s 2 sec s π 2 Γ s - 2 2 1 F 1 1 ; a 2 b 2 4 - s 2$

$a > 0 ; 0 < Re s < 3 ; | arg b | < π / 4$

$e - b 2 x 2 x 2 - a 2 erfi b x$

11

$π a s - 2 2 e - a 2 b 2 tan s π 2 erfi a b$

$+ b 2 - s 2 tan s π 2 Γ s - 2 2 1 F 1 1 ; - a 2 b 2 4 - s 2$

$a > 0 ; - 1 < Re s < 3 ; | arg b | < π / 4$

$erf φ x$

#### 3.6.5  erfc φ x $\text{ erf}\left(\varphi \left(x\right)\right)$ , f x $\text{ erfc}\left(\varphi \left(x\right)\right)$ , and algebraic functions

No.

$F s$

$a - x + α - 1$

1

$2 π a s + α b B 2 α + 1 2 , 2 s + 1 2 3 F 3 1 2 , 2 α + 1 2 , 2 s + 1 2 ; - a 2 b 2 4 3 2 , s + α + 1 2 , s + α + 2 2$

$× erf ( b x a - x )$

$a > 0 ; Re α , Re s > - 1 / 2$

$a - x + α - 1 erf b x a - x$

2

$2 π a s + α + 1 b B s + 1 , α + 1 5 F 5 1 2 , Δ 2 , s + 1 , Δ 2 , α + 1 3 2 , Δ 4 , s + α + 2 ; - a 4 b 2 16$

$a > 0 ; Re α , Re s > - 1$

$θ 1 - x erfc a x + b 1 - x 2$

3

$2 π e ( a 2 - b 2 ) / 2 Γ s D - s 2 a D - s - 1 2 b$

$Re s , Re b > 0$

$θ x - a erf bx x 2 - c 2$

4

$- 2 a s b π s Ψ 1 1 2 , - s 2 ; 2 - s 2 , 3 2 ; c 2 a 2 , - b 2$

$a > 0 ; Re s < 0 ; | c | < a$

$1 x + a ρ erf bx x + a$

5

$2 a s - ρ b π B s + 1 , ρ - s 3 F 3 1 2 , s + 1 2 , s + 2 2 3 2 , ρ + 1 2 , ρ + 2 2 ; - b 2$

$- 1 < Re s < Re ρ ; | arg a | < π$

$1 x 2 + a 2 ρ erf bx x 2 + a 2$

6

$a s - 2 ρ - 1 b π B s + 1 2 , 1 - s + 2 ρ 2 3 F 3 1 2 , s + 1 2 , 1 - s + 2 ρ 2 3 2 , ρ + 1 2 , ρ + 2 2 ; - b 2 4 a 2$

$Re a > 0 ; - 1 < Re s < 2 Re ρ + 1$

$erf φ x$

#### 3.6.6  erfc φ x $\text{ erf}\left(\varphi \left(x\right)\right)$ , f x $\text{ erfc}\left(\varphi \left(x\right)\right)$ , and the exponential function

No.

$F s$

$a - x + α - 1 e b 2 x ( a - x )$

1

$2 π a s + α b B s + 1 2 , α + 1 2 3 F 3 1 , 2 s + 1 2 , 2 α + 1 2 ; a 2 b 2 4 3 2 , s + α + 1 2 , s + α + 2 2$

$× erf ( b x ( a - x ) )$

$a > 0 ; Re α , Re s > - 1 / 2$

$a - x + α - 1 e b 2 x 2 a - x 2$

2

$2 π a s + α + 1 b B s + 1 , α + 1 6 F 5 1 , Δ 2 , s + 1 , Δ 2 , α + 1 3 2 , Δ 4 , s + α + 2 ; a 4 b 2 16$

$× erf b x ( a - x )$

$a > 0 ; Re s , Re α > - 1$

$θ x - a x 2 - b 2 e a 2 x 2 / x 2 - b 2$

3

$2 a s - 1 c π 1 - s Ψ 1 1 , 1 - s 2 ; 3 - s 2 , 3 2 ; c 2 a 2 , - b 2$

$× erf cx x 2 - c 2$

$a > 0 ; Re s < 0 ; | c | < a$

$1 x + a ρ e b 2 x 2 / x + a 2$

4

$2 a s - ρ b π B s + 1 , ρ - s 3 F 3 1 , s + 1 2 , s + 2 2 3 2 , ρ + 1 2 , ρ + 2 2 ; b 2$

$× erf bx x + a$

$- 1 < Re s < Re ρ ; | arg a | < π$

$1 x 2 + a 2 ρ e b 2 x 2 / x 2 + a 2 2$

5

$a s - 2 ρ - 1 b 2 π B s + 1 2 , 1 - s + 2 ρ 2 3 F 3 1 , s + 1 2 , 1 - s + 2 ρ 2 3 2 , ρ + 1 2 , ρ + 2 2 ; b 2 4 a 2$

$× erf bx x 2 + a 2$

$Re a > 0 ; - 1 < Re s < 2 Re ρ + 1$

$erf b x$

#### 3.6.7  erfc b x $\text{ erf}{\left(bx\right)}$ , δ = 1 0 $\text{ erfc}{\left(bx\right)}$ , and trigonometric functions

Notation: $f x$

.

No.

$F s$

$sin a x cos a x erf b x$

1

$- a δ b - s - δ π s + δ Γ s + δ + 1 2 2 F 2 s + δ 2 , s + δ + 1 2 ; - a 2 4 b 2 2 δ + 1 2 , s + δ + 2 2$

$+ Γ s a s sin s π / 2 cos s π / 2$

$a > 0 ; - δ - 1 < Re s < 1 ; | arg b | < π / 4$

$sin a x 2 cos a x 2 erf b x$

2

$- a δ b - s - 2 δ π s + 2 δ Γ s + 2 δ + 1 2 3 F 2 s + 2 δ 4 , s + 2 δ + 1 4 , s + 2 δ + 3 4 2 δ + 1 2 , s + 2 δ + 4 4 ; - a 2 b 4$

$+ a - s / 2 2 Γ s 2 sin s π / 4 cos s π / 4$

$a > 0 ; - 2 δ - 1 < Re s < 2 ; | arg b | < π / 4$

$sin a x 2 cos a x 2 erfc b x$

3

$a δ b - s - 2 δ π s + 2 δ Γ s + 2 δ + 1 2 3 F 2 s + 2 δ 4 , s + 2 δ + 1 4 , s + 2 δ + 3 4 2 δ + 1 2 , s + 2 δ + 4 4 ; - a 2 b 4$

$Re s > - 2 δ ; Re b 2 > | Im a |$

$sin a x erfc b x$

4

$2 a b - s - 1 / 2 π 2 s + 1 Γ 2 s + 3 4 2 F 4 2 s + 1 4 , 2 s + 3 4 ; a 4 256 b 2 1 2 , 3 4 , 5 4 , 2 s + 5 4$

$- a 3 b - s - 3 / 2 3 π 2 s + 3 Γ 2 s + 5 4 2 F 4 2 s + 3 4 , 2 s + 5 4 ; a 4 256 b 2 5 4 , 3 2 , 7 4 , 2 s + 7 4$

$Re s > - 1 / 2 ; | arg b | < π / 4$

$cos a x erfc b x$

5

$b - s π s Γ s + 1 2 2 F 4 s 2 , s + 1 2 ; a 4 256 b 2 1 4 , 1 2 , 3 4 , s + 2 2$

$- a 2 b - s - 1 2 π s + 1 Γ s + 2 2 2 F 4 s + 1 2 , s + 2 2 ; a 4 256 b 2 3 4 , 5 4 , 3 2 , s + 3 2$

$Re s > 0 ; | arg b | < π / 4$

$sin 2 n a x cos 2 n a x erfc b x$

6

$2 - 2 n b - s π s Γ s + 1 2 [ 2 ∑ k = 0 n - 1 ∓ 1 n - k 2 n k$

$× 2 F 2 s 2 , s + 1 2 ; - n - k 2 a 2 b 2 1 2 , s + 2 2 + 2 n n ]$

$a > 0 ; Re s > - 2 n δ ; | arg b | < π / 4 ; n ≥ 1$

$sin 2 n + 1 a x cos 2 n + 1 a x erfc b x$

7

$2 - 2 n a δ b - s - δ π s + δ Γ s + δ + 1 2 ∑ k = 0 n ( ∓ 1 ) n - k 2 n - 2 k + 1 δ$

$× 2 n + 1 k 2 F 2 s + δ 2 , s + δ + 1 2 ; - n - k + 1 2 2 a 2 b 2 2 δ + 1 2 , s + δ + 2 2$

$a > 0 ; Re s > - 2 n + 3 δ ; | arg b | < π / 4$

$sinh a x sin a x cosh a x cos a x$

8

$a 2 δ b - s - 2 δ π s + 2 δ Γ s + 2 δ + 1 2$

$× erfc b x$

$× 3 F 4 s + 2 δ 4 , s + 2 δ + 1 4 , s + 2 δ + 3 4 2 δ + 1 4 , 2 δ + 3 4 , 2 δ + 1 2 , s + 2 δ + 4 4 ; - a 4 16 b 4$

$a > 0 ; Re s > - 2 δ ; | arg b | < π / 4$

$sinh a x cos a x cosh a x sin a x$

9

$a b - s - 1 π s + 1 Γ s + 2 2 3 F 4 s + 1 4 , s + 2 4 , s + 4 4 1 2 , 3 4 , 5 4 , s + 5 4 ; - a 4 16 b 4$

$× erfc b x$

$∓ a 3 b - s - 1 3 π s + 3 Γ s + 4 2 3 F 4 s + 3 4 , s + 4 4 , s + 6 4 5 4 , 3 2 , 7 4 , s + 7 4 ; - a 4 16 b 4$

$a > 0 ; Re s > - 1 ; | arg b | < π / 4$

$erfc b x$

#### 3.6.8  erfi b x $\text{ erfc}{\left(bx\right)}$ , δ = 1 0 $\text{ erfi}{\left(bx\right)}$ , and the exponential or trigonometric functions

Notation: $f x$

.

No.

$F s$

$e - b 2 x 2 sin a x cos a x$

1

$∓ Γ s - 1 π a s - 1 b cos s π / 2 sin s π / 2 2 F 2 1 2 , 1 ; - a 2 4 b 2 2 - s 2 , 3 - s 2$

$× erfi b x$

$∓ a δ 2 b s + δ cot s π / 2 tan s π / 2 Γ s + δ 2 1 F 1 s + δ 2 ; - a 2 4 b 2 2 δ + 1 2$

$a > 0 ; - δ - 1 < Re s < 2 ; | arg b | < π / 4$

$e b 2 x 2 sin a x cos a x$

2

$∓ Γ s - 1 π a s - 1 b cos s π / 2 sin s π / 2 2 F 2 1 2 , 1 ; a 2 4 b 2 2 - s 2 , 3 - s 2$

$× erfc b x$

$∓ a δ 2 b s + δ csc s π / 2 sec s π / 2 Γ s + δ 2 1 F 1 s + δ 2 ; a 2 4 b 2 2 δ + 1 2$

$a > 0 ; - δ < Re s < 2 ; | arg b | < π / 4$

$e - b 2 x 2 sin a x 2 cos a x 2$

3

$1 2 π a ( s - 1 ) / 2 b sin s - 1 π / 4 cos s - 1 π / 4 Γ s - 1 2 3 F 2 1 4 , 3 4 , 1 ; - a 2 b 4 3 - s 4 , 5 - s 4$

$× erfi b x$

$- 1 4 π a ( s - 3 ) / 2 b 3 sin s + 1 π / 4 cos s + 1 π / 4$

$× Γ s - 3 2 3 F 2 3 4 , 1 , 5 4 ; - a 2 b 4 5 - s 4 , 7 - s 4$

$+ a δ 2 b s + 2 δ tan s π 2 Γ s + 2 δ 2 2 F 1 s + 2 δ 4 , s + 2 δ + 2 4 2 δ + 1 2 ; - a 2 b 4$

$a > 0 ; - 2 δ - 1 < Re s < 3 ; | arg b | < π / 4$

$e b 2 x 2 sin a x 2 erfc b x$

4

$- a ( 1 - s ) / 2 4 π b cos s π 2 csc s + 1 π 4 Γ s - 1 2 3 F 2 1 4 , 3 4 , 1 ; - a 2 b 4 3 - s 4 , 5 - s 4$

$+ a ( 3 - s ) / 2 8 π b 3 cos s π 2 sec s + 1 π 4 Γ s - 3 2 3 F 2 1 , 3 4 , 5 4 ; - a 2 b 4 5 - s 4 , 7 - s 4$

$- 1 2 a 2 + b 4 - s / 4 sec s π 2 sin s 2 arctan a b 2 Γ s 2$

$a > 0 ; - 2 < Re s < 3 ; | arg b | < π / 4$

$e b 2 x 2 cos a x 2 erfc b x$

5

$- a ( 1 - s ) / 2 4 π b cos s π 2 csc s - 1 π 4 Γ s - 1 2 3 F 2 1 4 , 3 4 , 1 ; - a 2 b 4 3 - s 4 , 5 - s 4$

$+ a ( 3 - s ) / 2 8 π b 3 cos s π 2 csc s + 1 π 4 Γ s - 3 2 3 F 2 1 , 3 4 , 5 4 , - a 2 b 4 5 - s 4 , 7 - s 4$

$+ 1 2 a 2 + b 4 - ( s + 2 ) / 4 sec s π 2 Γ s 2$

$× [ a sin s + 2 2 arctan a b 2 + b 2 cos s + 2 2 arctan a b 2 ]$

$a > 0 ; 0 < Re s < 3 ; | arg b | < π / 4$

$erf b x$

#### 3.6.9  erfc b x $\text{ erf}\left(bx\right)$ , f x $\text{ erfc}\left(bx\right)$ , and the logarithmic function

No.

$F s$

$ln x erf a x$

1

$a - s π s Γ s + 1 2 ln a + 1 s - 1 2 ψ s + 1 2$

$- 1 < Re s < 0 ; | arg a | < π / 4$

$ln x 2 + a 2 erf b x erfc b x$

2

$∓ a 2 b 2 - s π s Γ s - 1 2 2 F 2 1 , 1 ; a 2 b 2 2 , 3 - s 2$

$∓ 2 a 2 b 2 - s π s s - 2 Γ s - 1 2 2 F 2 1 , 2 - s 2 ; a 2 b 2 3 - s 2 , 4 - s 2$

$± b - s π s Γ s + 1 2 2 s - ψ s + 1 2 + 2 ln b$

$± π a s s erfi a b + π i 1 - s s b s γ s + 1 2 , - a 2 b 2 sec s π 2$

$+ 0 1 π a s s csc s π 2$

$Re a > 0 ; - 1 < Re s < 0 Re s > 0 ; | arg b | < π / 4$

$ln | x 2 - a 2 | erf b x erfc b x$

3

$± a 2 b 2 - s π s Γ s - 1 2 2 F 2 1 , 1 ; - a 2 b 2 2 , 3 - s 2$

$± 2 a 2 b 2 - s π s s - 2 Γ s - 1 2 2 F 2 1 , 2 - s 2 ; - a 2 b 2 3 - s 2 , 4 - s 2$

$± b - s π s Γ s + 1 2 2 s - ψ s + 1 2 + π tan s π 2 + 2 ln b$

$∓ π a s s erf a b + π b - s s Γ s + 1 2 , a 2 b 2 tan s π 2$

$+ 0 1 π a s s cot s π 2$

$a > 0 ; - 1 < Re s < 0 Re s > 0 ; | arg b | < π / 4$

$θ a - x ln a + a - x x$

4

$a s + 1 b s Γ s + 1 2 s + 3 2 3 F 3 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 3 2 , 2 s + 3 4 , 2 s + 5 4$

$× erf b x$

$- a s + 1 b s s + 1 Γ s + 1 2 s + 3 2 3 F 3 s + 1 2 , s + 1 2 , s + 2 2 ; - a 2 b 2 2 s + 3 4 , 2 s + 5 4 , s + 3 2$

$a > 0 ; Re s > - 1$

$θ a - x ln a + a - x x$

5

$a s + 1 b s + 1 Γ s + 1 2 s + 3 2 4 F 4 1 , s + 1 2 , s + 1 2 , s + 2 2 ; a 2 b 2 3 2 , 2 s + 3 4 , 2 s + 5 4 , s + 3 2$

$× e b 2 x 2 erf b x$

$a > 0 ; Re s > - 1$

$θ a - x ln a + a 2 - x 2 x$

6

$2 a s + 1 b s s + 1 Γ [ s + 1 2 s 2 ] 3 F 3 1 2 , s + 1 2 , s + 1 2 ; - a 2 b 2 3 2 , s + 2 2 , s + 3 2$

$× erf b x$

$a > 0 ; Re s > - 1$

$ln n x erf a x$

7

$- 1 π ∂ n ∂ s n a - s s Γ s + 1 2$

$- 1 < Re s < 0 ; | arg a | < π / 4$

$θ a - x ln n x a erf b x$

8

$2 - 1 n n ! a s + 1 b π s + 1 n + 1 n + 2 F n + 2 1 2 , s + 1 2 , ⋯ , s + 1 2 3 2 , s + 3 2 , ⋯ , s + 3 2 ; - a 2 b 2$

$a > 0 ; Re s > 0$

$θ a - x e b 2 x 2 ln n x a$

9

$2 - 1 n n ! a s + 1 b π s + 1 n + 1 n + 2 F n + 2 1 , s + 1 2 , ⋯ , s + 1 2 3 2 , s + 3 2 , ⋯ , s + 3 2 ; a 2 b 2$

$× erf b x$

$a > 0 ; Re s > - 1$

$erf a x$

#### 3.6.10  f x $\text{ erf}\left(ax\right)$ and inverse trigonometric functions

No.

$F s$

$θ 1 - x arcsin x arccos x$

1

$1 ± 1 π 4 s π erf a - a - s γ s + 1 2 , a 2$

$× erf a x$

$∓ a 2 s + 1 Γ [ s 2 s + 3 2 ] [ s + 1 2 F 2 1 2 , s + 2 2 ; - a 2 3 2 , s + 3 2$

$- 2 F 2 s + 1 2 , s + 2 2 ; - a 2 s + 3 2 , s + 3 2 ]$

$Re s > 0$

$θ a - x arccos x a$

2

$a s + 1 b 2 Γ [ s 2 s + 3 2 ] [ 2 F 2 1 2 , s + 2 2 3 2 , s + 3 2 ; - a 2 b 2$

$× erf b x$

$- 1 s + 1 2 F 2 s + 1 2 , s + 2 2 s + 3 2 , s + 3 2 ; - a 2 b 2 ]$

$a > 0 ; Re s > - 1$

$θ a - x e b 2 x 2 arccos x a$

3

$a s + 1 b s + 1 Γ [ s + 2 2 s + 3 2 ] 3 F 3 1 , s + 1 2 , s + 2 2 3 2 , s + 3 2 , s + 3 2 ; a 2 b 2$

$a > 0 ; Re s > - 1$

$× erf b x$

$arctan x erf a x$

4

$a 1 - s π s Γ s 2 [ 1 s - 1 2 F 2 1 , 1 - s 2 ; a 2 2 - s 2 , 3 - s 2 + 2 F 2 1 2 , 1 ; a 2 3 2 , 2 - s 2 ]$

$+ π 2 s csc s π 2 erfi a + π - a 2 ( 1 - s ) / 2 2 a s csc s π 2$

$× γ s + 1 2 , - a 2 - π a - s 2 s Γ s + 1 2$

$- 2 < Re s < 0 ; | arg a | < π / 4$

$erf b x$

#### 3.6.11  Ei - a x 2 $\text{ erf}{\left(bx\right)}$ and f x $\text{ Ei}\left(-ax^2\right)$

No.

$F s$

$Ei - a x 2 erf b x$

1

$- 2 a - ( s + 1 ) / 2 b π s + 1 Γ s + 1 2 3 F 2 1 2 , s + 1 2 , s + 1 2 3 2 , s + 3 2 ; - b 2 a$

$Re a > 0 ; Re s > - 1 ; | arg b | < π / 4$

$e b 2 x 2 Ei - a x 2 erf b x$

2

$- 2 a - ( s + 1 ) / 2 b π s + 1 Γ s + 1 2 3 F 2 1 , s + 1 2 , s + 1 2 3 2 , s + 3 2 ; b 2 a$

$Re a - b 2 > 0 ; Re s > - 1 ; | arg b | < π / 4$

$erf b x$

#### 3.6.12  erfc b x $\text{ erf}{\left(bx\right)}$ , si a x ${\text{ erfc}\left(bx\right)}$ , and ci a x $\text{ si}\left(ax\right)$ , Si a x ${\text{ ci}\left(ax\right)}$ , f x $\text{ Si}\left(ax\right)$

No.

$F s$

$si a x erf b x$

1

$a 3 b - s - 3 18 π s + 3 Γ s + 4 2 4 F 4 1 , 3 2 , s + 3 2 , s + 4 2 2 , 5 2 , 5 2 , s + 5 2 ; - a 2 4 b 2$

$- a b - s - 1 π s + 1 Γ s + 2 2 - a - s s sin s π 2 Γ s + π 2 b s s Γ s + 1 2$

$a > 0 ; - 1 < Re s < 2 ; | arg b | < π / 4$

$ci a x erf b x$

2

$a 2 b - s - 2 4 π s + 2 Γ s + 3 2 4 F 4 1 , 1 , s + 2 2 , s + 3 2 3 2 , 2 , 2 , s + 4 2 ; - a 2 4 b 2$

$+ b - s π s Γ s + 1 2 1 s - 1 2 ψ s + 1 2 + ln b a - C$

$- a - s s Γ s cos s π 2$

$a > 0 ; - 1 < Re s < 2 ; | arg b | < π / 4$

$Si a x erfc b x$

3

$a Γ s / 2 2 π b s + 1 [ 2 F 2 1 2 , s + 2 2 3 2 , 3 2 ; - a 2 4 b 2 - 1 s + 1 2 F 2 s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - a 2 4 b 2 ]$

$a > 0 ; Re s > - 1 ; | arg b | < π / 4$

$erf a x$

#### 3.6.13  Products of erfc b x $\text{ erf}\left(ax\right)$ , erfi c x $\text{ erfc}\left(bx\right)$ , f x $\text{ erfi}{\left(cx\right)}$

No.

$F s$

$erf a x erf b x erfc a x erfc b x$

1

$- 2 b π a s + 1 s + 1 Γ s + 2 2 3 F 2 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - b 2 a 2$

$∓ 1 π s b - s a - s Γ s + 1 2$

$- 2 < Re s < 0 Re s > 0 ; | arg a | , | arg b | < π / 4$

$erfi a x erfc a x$

2

$a - s π s tan s π 4 Γ s + 1 2$

$- 1 < Re s < 2 ; | arg a | < π / 4$

$erf a x erfc b x$

3

$2 b π a s + 1 ( s + 1 ) Γ s + 2 2 3 F 2 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - b 2 a 2$

$+ 1 π s b - s - a - s Γ s + 1 2$

$Re s > - 1 ; | arg a | , | arg b | < π / 4$

$1 - erf 2 a x$

4

$2 π a s Γ s 2 2 F 1 1 2 , s + 2 2 3 2 ; - 1$

$Re s > 0 ; | arg a | < π / 4$

$erf 2 a x$

5

$2 π 1 + s a s Γ s + 2 2 3 F 2 1 2 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - 1 - a - s π s Γ s + 1 2$

$- 2 < Re s < 0 ; | arg a | < π / 4$

$a - x + α - 1$

6

$4 π a s + α b 2 B 2 α + 1 2 , 2 s + 1 2 4 F 5 1 2 , 1 , 2 α + 1 2 , 2 s + 1 2 ; a 2 b 2 16 3 4 , 3 2 , 5 4 , s + α + 1 2 , s + α + 2 2$

$× erf ( b x a - x 4 )$

$a > 0 ; Re s , Re α > - 1 / 2$

$× erfi ( b x a - x 4 )$

$erfi a x erf a x erfc b x$

7

$4 a 2 b - s - 2 π 3 / 2 s + 2 Γ s + 3 2 5 F 4 1 2 , 1 , s + 2 4 , s + 3 4 , s + 4 4 3 4 , 5 4 , 3 2 , s + 6 4 ; a 4 4 b 4$

$Re b 2 - a 2 > 0 ; Re s > - 2 ; | arg a | , | arg b | < π / 4$

$erf a x$

#### 3.6.14  Products of erfc b x $\text{ erf}\left(ax\right)$ , erfi c x $\text{ erfc}\left(bx\right)$ , f x $\text{ erfi}{\left(cx\right)}$ , and algebraic functions

No.

$F s$

$a - x + α - 1$

1

$4 a s + α + 1 b 2 π B α , s + 2 6 F 7 1 2 , 1 , Δ 4 , s + 2 ; a 4 b 4 4 3 4 , 5 4 , 3 2 , Δ 4 , s + α + 2$

$× erf b x erfi b x$

$a , Re α > 0 ; Re s > - 2$

$a 2 - x 2 + α - 1$

2

$2 a s + 2 α b 2 π B α , s + 2 2 4 F 5 1 2 , 1 , s + 2 4 , s + 4 4 ; a 4 b 4 4 3 4 , 5 4 , 3 2 , s + 2 α + 2 4 , s + 2 α + 4 4$

$× erf b x erfi b x$

$a , Re α > 0 ; Re s > - 2$

$erf a x$

#### 3.6.15  Products of erfc b x $\text{ erf}\left(ax\right)$ , erfi c x $\text{ erfc}\left(bx\right)$ , f x $\text{ erfi}{\left(cx\right)}$ , and the exponential function

No.

$F s$

$e - a x 2 erfi b x erf b x$

1

$2 b 2 π a s / 2 + 1 Γ s + 2 2 4 F 3 1 2 , 1 , s + 2 4 , s + 4 4 3 2 , 3 4 , 5 4 ; b 4 a 2$

$Re a > Re b 2 ; Re s > - 2 ; | arg b | < π / 4$

$e - a 2 x 2 erfi a x erf b x$

2

$- 2 a π b s + 1 s + 1 Γ s + 2 2 3 F 2 1 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; - a 2 b 2$

$+ a - s 2 Γ s 2 tan s π 2$

$- 2 < Re s < 1 ; | arg a | , | arg b | < π / 4$

$e - a 2 + b 2 x 2 erfi a x$

3

$- b π a s + 1 cot s π 2 Γ s + 1 2 2 F 1 1 , s + 1 2 3 2 ; - b 2 a 2$

$× erfi b x$

$- b 1 - s 2 π a cot s π 2 Γ s - 1 2 2 F 1 1 2 , 1 3 - s 2 ; - b 2 a 2$

$| Re s | < 2 ; | arg a | , | arg b | < π / 4$

$e b 2 x 2 erfc a x erfc b x$

4

$- 2 b π a s + 1 s + 1 Γ s + 2 2 3 F 2 1 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; b 2 a 2$

$+ a - s s π Γ s + 1 2 2 F 1 s 2 , s + 1 2 s + 2 2 ; b 2 a 2$

$Re s > 0 ; | arg a | , | arg b | < π / 4$

$e a 2 x 2 erf a x erfc b x$

5

$2 a π b s + 1 s + 1 Γ s + 2 2 3 F 2 1 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; a 2 b 2$

$Re b 2 - a 2 > 0 ; Re s > - 1 ; | arg b | < π / 4$

$e b 2 x 2 erf a x erfc b x$

6

$2 b π a s + 1 s + 1 Γ s + 2 2 3 F 2 1 , s + 1 2 , s + 2 2 3 2 , s + 3 2 ; b 2 a 2$

$- 1 π a s s Γ s + 1 2 2 F 1 s 2 , s + 1 2 s + 2 2 ; b 2 a 2 + b - s 2 Γ s 2 sec s π 2$

$| Re s | < 1 ; | arg a | , | arg b | < π / 4$

$e - a x 4 erf b x erfi b x$

7

$b 2 π a ( s + 2 ) / 4 Γ s + 2 4 3 F 3 1 2 , 1 , s + 2 4 ; b 4 4 a 3 4 , 5 4 , 3 2$

$Re a > 0 ; Re s > - 2 ; | arg b | < π / 4$

$erf a x$

#### 3.6.16  Products of erfc b x $\text{ erf}\left(ax\right)$ , erfi c x $\text{ erfc}\left(bx\right)$ , f x $\text{ erfi}{\left(cx\right)}$ , and the logarithmic function

No.

$F s$

$θ a - x ln a - x + a x$

1

$2 a s + 2 b 2 π s + 2 Γ s + 2 2 s + 5 2 7 F 8 1 2 , 1 , s + 2 4 , Δ 4 , s + 4 2 ; a 4 b 4 4 3 4 , 5 4 , 3 2 , s + 6 4 , Δ 4 , s + 2 2$

$× erf b x erfi b x$

$a > 0 ; Re s > - 2$

$θ a - x ln a 2 - x 2 + a x$

2

$a s + 2 b 2 π Γ [ s 2 s + 3 2 ] [ 4 F 3 1 2 , 1 , s + 2 4 , s + 4 4 ; a 2 b 4 4 3 4 , 5 4 , 3 2 , s + 3 4 , s + 5 4$

$× erf b x erfi b x$

$- 2 s + 2 4 F 3 1 , s + 2 4 , s + 2 4 , s + 4 4 ; a 2 b 4 4 3 4 , 5 4 , s + 3 4 , s + 5 4 , s + 6 4 ]$

$a > 0 ; Re s > - 2$

$θ a - x ln n x a$

3

$4 - 1 n n ! a s + 2 b 2 π s + 2 n + 1 n + 3 F n + 4 1 2 , 1 , s + 2 4 , ⋯ , s + 2 4 ; a 4 b 4 4 3 4 , 5 4 , 3 2 , s + 8 4 , ⋯ , s + 8 4$

$× erf b x erfi b x$

$a > 0 ; Re s > - 2$

$erf a x$

#### 3.6.17  Products of erfc b x $\text{ erf}\left(ax\right)$ , erfi c x $\text{ erfc}\left(bx\right)$ , f x $\text{ erfi}{\left(cx\right)}$ , and inverse trigonometric functions

No.

$F s$

$θ a - x arccos x a$

1

$2 a s + 2 b 2 π s Γ [ s + 3 2 s + 4 2 ] [ 4 F 5 1 2 , 1 , s + 3 4 , s + 5 4 ; a 4 b 4 4 3 4 , 5 4 , 3 2 , s + 4 4 , s + 6 4$

$× erf b x erfi b x$

$- 2 s + 2 4 F 5 1 , s + 2 4 , s + 3 4 , s + 5 4 ; a 4 b 4 4 3 4 , 5 4 , s + 4 4 , s + 6 4 , s + 6 4 ]$

$a > 0 ; Re s > - 2$

$S z$

#### 3.7  The Fresnel Integrals C z ${ \text{ S}\left(z\right)}$ and S z C z = 1 ± i 4 erf 1 + i z 2 ∓ erfi 1 + i z 2 , S z C z = i 1 z { 1 2 2 i z [ 1 - e - i z π Ψ 1 2 , 1 2 , i z ] ∓ 1 2 - 2 i z [ 1 - e iz π Ψ 1 2 , 1 2 , - i z ] } , S z = 1 3 2 z 3 π 1 F 2 3 4 ; 3 2 , 7 4 ; - z 2 4 , C z = 2 z π 1 F 2 1 4 ; 1 2 , 5 4 ; - z 2 4 , S z = π z 3 / 8 2 - z 3 / 4 G 13 10 - z 2 4 | 1 3 / 4 , 1 / 4 , 0 , C z = π z 1 / 8 2 - z 1 / 4 G 13 10 - z 2 4 | 1 1 / 4 , 3 / 4 , 0 , S ( z 2 ) = 1 2 - 1 2 G 13 20 z 2 4 | 1 0 , 3 / 4 , 1 / 4 , C ( z 2 ) = 1 2 - 1 2 G 13 20 z 2 4 | 1 0 , 1 / 4 , 3 / 4 , S 2 ( z 2 ) + C 2 ( z 2 ) = 1 2 G 24 12 z 2 4 | 1 / 2 , 1 1 / 2 , 3 / 4 , 1 / 4 , 0 . ${ \text{ C}\left(z\right)}$

More formulas can be obtained from the corresponding sections due to the relations

$S φ x$

#### 3.7.1  C φ x $\text{ S}\left(\varphi \left(x\right)\right)$ , δ = 1 0 $\text{ C}\left(\varphi \left(x\right)\right)$ , and algebraic functions

Notation: $f x$

.

No.

$F s$

$S a x C a x$

1

$- a - s 2 π s Γ 2 s + 1 2 sin 2 s + 1 π / 4 cos 2 s + 1 π / 4$

$a > 0 ; - 1 ∓ 1 / 2 < Re s < 0$

$1 2 - S a x C a x$

2

$a - s 2 π s Γ 2 s + 1 2 sin 2 s + 1 π / 4 cos 2 s + 1 π / 4$

$a > 0 ; 0 < Re s < 3 / 2$

$a - x + α - 1 S b x C b x$