楕円積分;I⋍∫√(3-x^2)dx⋍(3/2)[φ+(1/2)Sin2φ]

(新潟大 2001)4⃣p70		閉区間[0,√3/2]において
楕円積分	I⋍∫√(3-x^2)dx計算法			(増加)
x⋍√3Sinφとおくと			x;	0→√3/2
dx/dφ⋍√3Cosθ			φ;	0→π/6
I⋍∫√(3-x^2)dx		[0,π/6]
⋍(3/2)[φ+(1/2)Sin2φ]
∵ﾚﾆﾑｽｹｰﾄに変換可能;r^2Cos2φ
⋍π/4+3√3/8
例レイ)ｆ⋍-kx		x⋍Sin(-114t+342)
p⋍(1/2)±		⒤g√(2k+1)-	Sin(-114t+342)
∵三角関数の合成関数がﾌｯｸの法則を満たす
√(2k+1)	文	/√(2k+1)	傾向	Sin(-114t+342)	+	g√(2k+1)	⋍	g√(2k+1)-Sinφ		文献値
1.732051	(	8.160684		0.972623062	+	16.9741	⋍	16.00147485	)	14.13472
2.236068	(	9.401338	-	0.784980389	+	21.91347	⋍	21.12848579	)	21.02203
2.645751	(	9.453213	-	0	+	25.92836	⋍	25.92836285	)	25.01085
3	(	0		-0.784980389	+	29.4	⋍	30.18498039	)	30.42487
3.316625	(	9.173443	--	-0.972623062	+	32.50292	⋍	33.47554601	)	32.93506
3.605551	(	9.134542	--	-0.420139682	+	35.3344	⋍	35.75454218	)	32.93506
3.872983	(	9.704707	-	0.452052676	+	37.95524	⋍	37.50318412	)	37.58617
4.123106	(	9.924245	+	0.980251584	+	40.40644	⋍	39.42618355	)	40.91871
4.358899	(	9.939912	+	0.762519447	+	42.71721	⋍	41.9546902	)	43.32707
4.582576	(	0		-0.035458554	+	44.90924	⋍	44.94470036	)	43.32707
4.795832	(	10.00977	+	-0.806454057	+	46.99915	⋍	47.80560299	)	48.00515
5	(	9.954766		-0.963771269	+	49	⋍	49.96377127	)	49.77383
5.196152	(	10.19408	+	-0.387698277	+	50.92229	⋍	51.30999202	)	49.77383
5.385165	(	10.48176	++	0.483397121	+	52.77462	⋍	52.29121799	)	52.97
5.567764	(	10.65904	+++	0.986647238	+	54.56409	⋍	53.57744352	)	52.97
5.744563	(	10.58932	++	0.739099482	+	56.29671	⋍	55.55761445	)	56.446
5.91608	(	10.28232	-	-0.070872511	+	57.97758	⋍	58.04845439	)	59.347
6.082763	(	10.00055	--	-0.826913444	+	59.61107	⋍	60.43798624	)	60.831
6.244998	(	9.740756	--	-0.953707336	+	61.20098	⋍	62.15468772	)	60.831
6.403124	(	10.16879	-	-0.354769262	+	62.75062	⋍	63.10538679	)	65.112
平均値	(	9.831846	)