フーリエ級数の公式
(π/2)sinh (π-x)＝sinh(π)｛1sinx/(1^2+1) + 2sin2x/(2^2+1) + 3sin3x/(3^2+1) + ・・｝
( 0 < x < 2π)
に作用素∫(0～p) e^(-x)を作用させて次式を導出した。
[p＝2π]
● 1/(1^2+1)^2 + 1/(2^2+1)^2 + 1/(3^2+1)^2 + 1/(4^2+1)^2 + ・・・
＝-1/2 + (π/4)(e^(2π)+1)/(e^(2π)-1) + π^2・e^(2π)/(e^(2π)-1)^2
または∑記号で表現すると次となる。
●∑(n＝1～∞) 1/(n^2+1)^2＝-1/2 + (π/4)(e^(2π)+1)/(e^(2π)-1) + π^2・e^(2π)/(e^(2π)-1)^2
[p＝π]
● 1/(1^2+1)^2 + 1/(3^2+1)^2 + 1/(5^2+1)^2 + ・・・＝π/8 - (π/4)/(e^π+1) - (π^2/4)e^π/(e^π+1)^2
● 1/(2^2+1)^2 + 1/(4^2+1)^2 + 1/(6^2+1)^2 + ・・・＝-1/2 + π/8 + (π/4)/(e^π-1) + (π^2/4)e^π/(e^π-1)^2
または∑記号で表現すると次となる。
●∑(n＝1～∞) 1/((2n-1)^2+1)^2＝π/8 - (π/4)/(e^π+1) - (π^2/4)e^π/(e^π+1)^2
●∑(n＝1～∞) 1/((2n)^2+1)^2＝-1/2 + π/8 + (π/4)/(e^π-1) + (π^2/4)e^π/(e^π-1)^2
[p＝π/2，3π/2]
● 1/(1^2+1)^2 - 3/(3^2+1)^2 + 5/(5^2+1)^2 - 7/(7^2+1)^2 + ・・・
＝(π^2/2)｛e^(7π/2)-e^(5π/2)｝/(e^(2π)-1)^2 - (π/8)｛3πe^(3π/2)-πe^(π/2)｝/(e^(2π)-1)
● 1/(2^2+1)^2 - 1/(4^2+1)^2 + 1/(6^2+1)^2 - 1/(8^2+1)^2 + ・・・
＝1/2 - (π^2/2)｛e^(7π/2)+e^(5π/2)｝/(e^(2π)-1)^2 + (π/8)｛(3π-2)e^(3π/2)+(π-2)e^(π/2)｝/(e^(2π)-1)
∑記号で表現すると次となる。
●∑(n＝1～∞) 1/((2n-1)^2+1)^2
＝(π^2/2)｛e^(7π/2)-e^(5π/2)｝/(e^(2π)-1)^2 - (π/8)｛3π・e^(3π/2)-πe^(π/2)｝/(e^(2π)-1)
●∑(n＝1～∞) 1/((2n)^2+1)^2
＝1/2 - (π^2/2)｛e^(7π/2)+e^(5π/2)｝/(e^(2π)-1)^2 + (π/8)｛(3π-2)e^(3π/2)+(π-2)e^(π/2)｝/(e^(2π)-1)
[p＝π/3，5π/3]
● 1/(1^2+1)^2 + 2/(2^2+1)^2 - 4/(4^2+1)^2 - 5/(5^2+1)^2
+ 7/(7^2+1)^2 + 8/(8^2+1)^2 - 10/(10^2+1)^2 - 11/(11^2+1)^2 + ・・・
＝ (π^2/√3)｛e^(11π/3)-e^(7π/3)｝/(e^(2π)-1)^2 - (π^2/(6√3))｛5e^(5π/3)-e^(π/3) ｝/(e^(2π)-1)
● 1/(3^2+1)^2 - 1/(6^2+1)^2 + 1/(9^2+1)^2 - 1/(12^2+1)^2 + ・・・
＝1/2 - (π/6)｛e^(5π/3)+e^π+e^(π/3)｝/(e^(2π)-1) + (π^2/18)｛5e^(5π/3)+e^(π/3)｝/(e^(2π)-1)
- (π^2/6)｛2e^(11π/3)+e^(3π)+2e^(7π/3)+e^π｝/(e^(2π)-1)^2
∑記号で表現すると次となる。
● ∑(n＝1～∞) (-1)^(n+1)｛(3n-2)/((3n-2)^2+1)^2 + (3n-1)/((3n-1)^2+1)^2｝
＝ (π^2/√3)｛e^(11π/3)-e^(7π/3)｝/(e^(2π)-1)^2 - (π^2/(6√3))｛5e^(5π/3)-e^(π/3) ｝/(e^(2π)-1)
● ∑(n＝1～∞) (-1)^(n+1)/((3n)^2+1)^2
＝1/2 - (π/6)｛e^(5π/3)+e^π+e^(π/3)｝/(e^(2π)-1) + (π^2/18)｛5e^(5π/3)+e^(π/3)｝/(e^(2π)-1)
- (π^2/6)｛2e^(11π/3)+e^(3π)+2e^(7π/3)+e^π｝/(e^(2π)-1)^2
[p＝π/4，3π/4，5π/4，7π/4]
● 1/(1^2+1)^2 + 3/(3^2+1)^2 - 5/(5^2+1)^2 - 7/(7^2+1)^2
+ 9/(9^2+1)^2 + 11/(11^2+1)^2 - 13/(13^2+1)^2 - 15/(15^2+1)^2 + ・・・
＝ (π^2/(2√2))｛e^(15π/4)+e^(13π/4)-e^(11π/4)-e^(9π/4)｝/(e^(2π)-1)^2
+ (π^2/(16√2))｛3e^(3π/4)+e^(π/4)-7e^(7π/4) - 5e^(5π/4)｝/(e^(2π)-1)
● 2/(2^2+1)^2 - 6/(6^2+1)^2 + 10/(10^2+1)^2 - 14/(14^2+1)^2 + ・・・
＝ (π^2/4)｛e^(15π/4)+e^(11π/4)-e^(13π/4)-e^(9π/4)｝/(e^(2π)-1)^2
+ (π^2/32)｛5e^(5π/4)+e^(π/4)-7e^(7π/4)-3e^(3π/4)｝/(e^(2π)-1)
● 1/(1^2+1)^2 - 1/(3^2+1)^2 - 1/(5^2+1)^2 + 1/(7^2+1)^2
+ 1/(9^2+1)^2 - 1/(11^2+1)^2 - 1/(13^2+1)^2 + 1/(15^2+1)^2 + ・・・
＝ (π/(4√2))｛e^(7π/4)+e^(π/4)-e^(5π/4)-e^(3π/4)｝/(e^(2π)-1)
+ (π^2/(2√2))｛e^(15π/4)+e^(9π/4)-e^(13π/4)-e^(11π/4)｝/(e^(2π)-1)^2
+ (π^2/(16√2))｛3e^(3π/4)+5e^(5π/4)-7e^(7π/4)-e^(π/4)｝/(e^(2π)-1)
● 1/(4^2+1)^2 - 1/(8^2+1)^2 + 1/(12^2+1)^2 - 1/(16^2+1)^2 + ・・・
＝1/2 - (π/8)｛e^(7π/4)+e^(5π/4)+e^(3π/4)+e^(π/4)｝/(e^(2π)-1)
+ (π^2/32)｛7e^(7π/4)+5e^(5π/4)+3e^(3π/4)+e^(π/4)｝/(e^(2π)-1)
- (π^2/4)｛e^(15π/4)+e^(13π/4)+e^(11π/4)+e^(9/4)｝/(e^(2π)-1)^2
∑記号で表現すると次となる。
● ∑(n＝1～∞) (-1)^(n+1)｛(4n-3)/((4n-3)^2+1)^2 + (4n-1)/((4n-1)^2+1)^2｝
＝ (π^2/(2√2))｛e^(15π/4)+e^(13π/4)-e^(11π/4)-e^(9π/4)｝/(e^(2π)-1)^2
+ (π^2/(16√2))｛3e^(3π/4)+e^(π/4)-7e^(7π/4) - 5e^(5π/4)｝/(e^(2π)-1)
● ∑(n＝1～∞) (-1)^(n+1)・(4n-2)/((4n-2)^2+1)^2
＝ (π^2/4)｛e^(15π/4)+e^(11π/4)-e^(13π/4)-e^(9π/4)｝/(e^(2π)-1)^2
+ (π^2/32)｛5e^(5π/4)+e^(π/4)-7e^(7π/4)-3e^(3π/4)｝/(e^(2π)-1)
● ∑(n＝1～∞) (-1)^(n+1)｛1/((4n-3)^2+1)^2 - 1/((4n-1)^2+1)^2｝
＝(π/(4√2))｛e^(7π/4)+e^(π/4)-e^(5π/4)-e^(3π/4)｝/(e^(2π)-1)
+ (π^2/(2√2))｛e^(15π/4)+e^(9π/4)-e^(13π/4)-e^(11π/4)｝/(e^(2π)-1)^2
+ (π^2/(16√2))｛3e^(3π/4)+5e^(5π/4)-7e^(7π/4)-e^(π/4)｝/(e^(2π)-1)
● ∑(n＝1～∞) (-1)^(n+1)/((4n)^2+1)^2
＝1/2 - (π/8)｛e^(7π/4)+e^(5π/4)+e^(3π/4)+e^(π/4)｝/(e^(2π)-1)
+ (π^2/32)｛7e^(7π/4)+5e^(5π/4)+3e^(3π/4)+e^(π/4)｝/(e^(2π)-1)
- (π^2/4)｛e^(15π/4)+e^(13π/4)+e^(11π/4)+e^(9/4)｝/(e^(2π)-1)^2
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