APPLICATIONS OF WALLIS THEOREM Mihály Bencze1, Florentin Smarandache2 1
Department of Mathematics, Áprily Lajos College, Braşov, Romania; Chair of Department of Math & Sciences, University of New Mexico, Gallup, NM 87301, USA
2
Abstract: In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals. Let’s recall Wallis Theorem: Theorem 1. (Wallis, 1616-1703) π
π
2 ⋅ 4 ⋅ ...⋅ (2n) . 1⋅ 3⋅ ...⋅ (2n + 1) 0 0 Proof: Using the integration by parts, we obtain 2
2
2n +1 2n+1 ∫ sin xdx = ∫ cos xdx =
π
π
2
I n = ∫ sin
2n +1
π
2
xdx = ∫ sin x sin xdx = − cos x ⋅ sin 2nx 2n
0
2
0
+
0
π 2
(
)
+2n ∫ sin 2n +1 x 1 − sin 2 x dx = 2nI n−1 − 2nI n 0
from where:
2n I n −1 . 2n + 1 By multiplication, we obtain the statement. We prove in the same manner for cos x . In =
Theorem 2. π
π
2
2
0
0
2n 2n ∫ sin xdx = ∫ cos xdx =
1 ⋅ 3 ⋅ ... ⋅ (2n − 1) π ⋅ . 2 ⋅ 4 ⋅ ... ⋅ (2n) 2
Proof: Same as the first theorem. ∞
Theorem 3. If f (x) = ∑ a2 k x 2 k , then k=0
π 2
∫ 0
π 2
f (sin x)dx = ∫ f (cos x)dx = 0
1
π 2
a0 +
π
∞
∑a 2
2k
k =1
1 ⋅ 3 ⋅ ... ⋅ (2k − 1) . 2 ⋅ 4 ⋅ ... ⋅ (2k)