Question Number 218560 by MrGaster last updated on 12/Apr/25
Commented by MrGaster last updated on 12/Apr/25
![J_0 (a(√(1−u^2 )))=Σ_(n=0) ^∞ (((−1)^n a^(2n) )/((n!)^2 2^(2n) ))(1−u^2 )^n ⇒Σ_(m=0) ^∞ (((−1)^n a^(2n) )/((n!)^2 2^(2n) ))∫_(cos α) ^1 (1−u^2 )^n (1+u)e^(ilu) du =Σ_(n=0) ^∞ (((−1)^m a^(2n) )/((n!)^2 2^(2n) ))∫_(cos α) ^1 (1−u)^n (1+u)^(u+1) e^(idu) du Handle internal integration: ∫_(cos α) ^1 (1−u)^n (1+u)^(n+1) e^(idu) du =Σ_(k=0) ^∞ (((ib)^k )/(k!))∫_(cos α) ^1 (1−u)^n (1+u)^(n+1) u^k du =Σ_(k=0) ^∞ (((ib)^k )/(k!))∫_(cos α) ^1 (1−u)^n Σ_(m=0) ^(n+1) C_(n+1) ^m u^(k+m) du =Σ_(λ=0) ^∞ Σ_(m=0) ^(n+1) C_(n+1) ^m (((ib)^k )/(k!))∫_(cos α) ^1 (1−u)^n u^(k+m) du =Σ_(k=0) ^∞ Σ_(m=0) ^(n+1) C_(n+1) ^m (((ib)^k )/(k!))[B(k+m+1,n+1)−B(cos α;k+m+1,n+1)] =Σ_(k=0) ^∞ Σ_(m=0) ^(n+1) C_(n+1) ^m (((−ib)^k )/(k!)) (((1−cos α)^(k+m+n+1) )/(k+m+n+1)) _2 F_1 (−n,k+m+1;k+m+n+2;1−cos α) Back to the original formula. .…… Quadruple series. determinant (((Σ_(n=0) ^∞ Σ_(k=0) ^∞ Σ_(m=0) ^(n+1) (((−1)^n a^(2n) )/((n!)^2 2^(2n) )) (((−ib)^k )/(k!)) (((1−cos α)^(k+m+n+1) )/(k+m+n+1))C_(n+1) ^m )),(( _2 F_1 (−n,k+m+1;k+m+n+2;1−cos α)))) ……… I dont know how thee convrgence is but I dontt wan to simplify.](Q218561.png)
$${J}_{\mathrm{0}} \left({a}\sqrt{\mathrm{1}−{u}^{\mathrm{2}} }\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {a}^{\mathrm{2}{n}} }{\left({n}!\right)^{\mathrm{2}} \mathrm{2}^{\mathrm{2}{n}} }\left(\mathrm{1}−{u}^{\mathrm{2}} \right)^{{n}} \\ $$$$\Rightarrow\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {a}^{\mathrm{2}{n}} }{\left({n}!\right)^{\mathrm{2}} \mathrm{2}^{\mathrm{2}{n}} }\int_{\mathrm{cos}\:\alpha} ^{\mathrm{1}} \left(\mathrm{1}−{u}^{\mathrm{2}} \right)^{{n}} \left(\mathrm{1}+{u}\right){e}^{{ilu}} {du} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{m}} {a}^{\mathrm{2}{n}} }{\left({n}!\right)^{\mathrm{2}} \mathrm{2}^{\mathrm{2}{n}} }\int_{\mathrm{cos}\:\alpha} ^{\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} \left(\mathrm{1}+{u}\right)^{{u}+\mathrm{1}} {e}^{{idu}} {du} \\ $$$$\mathrm{Handle}\:\mathrm{internal}\:\mathrm{integration}: \\ $$$$\int_{\mathrm{cos}\:\alpha} ^{\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} \left(\mathrm{1}+{u}\right)^{{n}+\mathrm{1}} {e}^{{idu}} {du} \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({ib}\right)^{{k}} }{{k}!}\int_{\mathrm{cos}\:\alpha} ^{\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} \left(\mathrm{1}+{u}\right)^{{n}+\mathrm{1}} {u}^{{k}} {du} \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({ib}\right)^{{k}} }{{k}!}\int_{\mathrm{cos}\:\alpha} ^{\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} \underset{{m}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\sum}}{C}_{{n}+\mathrm{1}} ^{{m}} {u}^{{k}+{m}} {du} \\ $$$$=\underset{\lambda=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{m}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\sum}}{C}_{{n}+\mathrm{1}} ^{{m}} \frac{\left({ib}\right)^{{k}} }{{k}!}\int_{\mathrm{cos}\:\alpha} ^{\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} {u}^{{k}+{m}} {du} \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{m}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\sum}}{C}_{{n}+\mathrm{1}} ^{{m}} \frac{\left({ib}\right)^{{k}} }{{k}!}\left[{B}\left({k}+{m}+\mathrm{1},{n}+\mathrm{1}\right)−{B}\left(\mathrm{cos}\:\alpha;{k}+{m}+\mathrm{1},{n}+\mathrm{1}\right)\right] \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{m}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\sum}}{C}_{{n}+\mathrm{1}} ^{{m}} \frac{\left(−{ib}\right)^{{k}} }{{k}!}\:\frac{\left(\mathrm{1}−\mathrm{cos}\:\alpha\right)^{{k}+{m}+{n}+\mathrm{1}} }{{k}+{m}+{n}+\mathrm{1}} \\ $$$$\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(−{n},{k}+{m}+\mathrm{1};{k}+{m}+{n}+\mathrm{2};\mathrm{1}−\mathrm{cos}\:\alpha\right) \\ $$$$\mathrm{Back}\:\mathrm{to}\:\mathrm{the}\:\mathrm{original}\:\mathrm{formula}. \\ $$$$.\ldots\ldots\:\mathrm{Quadruple}\:\mathrm{series}. \\ $$$$\begin{array}{|c|c|}{\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{m}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {a}^{\mathrm{2}{n}} }{\left({n}!\right)^{\mathrm{2}} \mathrm{2}^{\mathrm{2}{n}} }\:\frac{\left(−{ib}\right)^{{k}} }{{k}!}\:\frac{\left(\mathrm{1}−\mathrm{cos}\:\alpha\right)^{{k}+{m}+{n}+\mathrm{1}} }{{k}+{m}+{n}+\mathrm{1}}{C}_{{n}+\mathrm{1}} ^{{m}} }\\{\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(−{n},{k}+{m}+\mathrm{1};{k}+{m}+{n}+\mathrm{2};\mathrm{1}−\mathrm{cos}\:\alpha\right)}\\\hline\end{array} \\ $$$$\ldots\ldots\ldots\:\mathrm{I}\:\mathrm{dont}\:\mathrm{know}\:\mathrm{how}\:\mathrm{thee} \\ $$$$\mathrm{convrgence}\:\mathrm{is}\:\mathrm{but}\:\mathrm{I}\:\mathrm{dontt} \\ $$$$\mathrm{wan}\:\mathrm{to}\:\mathrm{simplify}. \\ $$