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USEFUL DATA
521
Advanced Integrals and Derivatives
∫
∂
∂a
a
∂
∂a
f (x ) ⋅ dx = f (a )
b
J 0 (x ) =
2
π
∫
π
cos(x ⋅ sin [θ ]) ⋅ dθ =
2
0
d
E (k )
K (k )
⋅ Κ (k ) =
−
2
dk
k ⋅ 1− k
k
(
)
d
1
⋅ E (k ) = [E (k ) − K (k )]
dk
k
2
π
∫
∫
b
f (x ) ⋅ dx = − f (a )
(where b is not a function of a)
a
π
cos(x ⋅ cos[θ ]) ⋅ dθ
2
(Zero order Bessel function)
0
(derivative of complete elliptic integral of the first kind)
(derivative of complete elliptic integral of the second kind)
Many problems in electrostatics and electromagnetics give rise to integrals which do not have
solutions in terms of elementary functions. The following set of definite integrals have solutions in
terms of the standard elliptic integrals.
∫
∫
∫
π
dφ
a ± b ⋅ cos(φ )
0
π
=
dφ
(a ± b ⋅ cos(φ ))3
0
π
∫
π
-
=
a ± b ⋅ sin (φ )
π
2
∫
a ± b ⋅ cos(φ ) dφ =
0
dφ
2
π
dφ
2
-
∫
π
2
π
2
-
π
2
 2b
K
a + b  a + b
2
=
(a ± b ⋅ sin(φ ))3
=




 2b
E
(a − b) a + b  a + b
2
 2b
a ± b ⋅ sin (φ ) dφ = 2 a + b E 
 a+b









a >b>0
Hypergeometric Functions
Although other hypergeometric functions do exist, unless otherwise specified assume that the Gauss
Hypergeometric Function is being referred to. It is defined by:
F (a, b; c; z ) ≡ 2 F1 (a, b; c; z ) ≡ 1 +
a ⋅ b z a(a + 1) b (b + 1) z 2 a(a + 1)(a + 2) b(b + 1)(b + 2) z 3
⋅ +
⋅
+
⋅
+K
c 1!
c (c + 1)
2!
c (c + 1)(c + 2)
3!
This hypergeometric function is useful for computing difficult functions such as complete elliptic
integrals, inverse trig functions &c. For example:
1 1 3

arcsin (x ) = x ⋅ F  , ; ; x 2 
2 2 2


3 1
2 2
π = 2 ⋅ F 1, 1; ; 

1+ n 1− n 3 2 
sin (n ⋅ arcsin (x )) = nx ⋅ F 
,
; ;x 
2 2
 2

K (k ) =
π
E (k ) =
π
1 1

⋅ F  , ; 1; k 2 
2
2 2

 1 1

⋅ F  − , ; 1; k 2 
2
2
2


(complete elliptic integral of the first kind)
(complete elliptic integral of the second kind)