Pagina:Werkecarlf03gausrich.djvu/141

I. $(t+u)^{n}=t^{u}F(-n,\beta ,\beta ,-{\tfrac {u}{t}})$
ubi elementum $\beta$ est arbitrarium.

II. $(t+u)^{n}+(t-u)^{n}=2t^{n}F(-{\tfrac {1}{2}}n,-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {uu}{tt}})$

III. $(t+u)^{n}+t^{n}=2t^{n}F(-n,\omega ,2\omega ,-{\tfrac {u}{t}})$
denotante $\omega$ quantitatem infinite parvam.

IV. $(t+u)^{n}-(t-u)^{n}=2nt^{n-1}uF(-{\tfrac {1}{2}}n+{\tfrac {1}{2}},-{\tfrac {1}{2}}n+1,{\tfrac {3}{2}},{\tfrac {uu}{tt}})$

V. $(t+u)^{n}-t^{n}=nt^{n-1}uF(1-n,1,2,-{\tfrac {u}{t}})$

VI. $\log(1+t)=tF(1,1,2,-t)$

VII. $\log {\tfrac {1+t}{1-t}}=2tF({\tfrac {1}{2}},1,{\tfrac {3}{2}},tt)$

VIII. $e^{t}=F(1,k,1,{\tfrac {t}{k}})=1+tF(1,k,2,{\tfrac {t}{k}})=1+t+{\tfrac {1}{2}}ttF(1,k,3,{\tfrac {t}{k}})$ etc.
denotante $e$ basin logarithmorum hyperbolicorum, $k$ numerum infinite magnum.

IX. $e^{t}+e^{-t}=2{F}(k,k^{\prime },{\tfrac {1}{2}},{\tfrac {tt}{4kk^{\prime }}})$
denotantibus $k,k^{\prime }$ numeros infinite magnos.

X. $e^{t}-e^{-t}=2tF(k,k^{\prime },{\tfrac {3}{2}},{\tfrac {tt}{4kk^{\prime }}})$

XI. $\sin t=tF(k,k^{\prime },{\tfrac {3}{2}},-{\tfrac {tt}{4kk^{\prime }}})$

XII. $\cos t=F(k,k^{\prime },{\tfrac {1}{2}},-{\tfrac {tt}{4kk^{\prime }}})$

XIII. $t=\sin t.F({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}},{\sin t}^{2})$

XIV. $t=\sin t.\cos t.F(1,1,{\tfrac {3}{2}},{\sin t}^{2})$

XV. $t=\operatorname {tang} t.F({\tfrac {1}{2}},1,{\tfrac {3}{2}},-{\operatorname {tang} t}^{2})$

XVI. $\sin nt=n\sin t.F({\tfrac {1}{2}}n+{\tfrac {1}{2}},-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {3}{2}},{\sin t}^{2})$

XVII. $\sin nt=n\sin t.\cos t.F({\tfrac {1}{2}}n+1,-{\tfrac {1}{2}}n+1,{\tfrac {3}{2}},{\sin t}^{2})$

XVIII. $\sin nt=n\sin t.{\cos t}^{n-1}F(-{\tfrac {1}{2}}n+1,-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {3}{2}},-{\operatorname {tang} t}^{2})$

XIX. $\sin nt=n\sin t.{\cos t}^{-n-1}F({\tfrac {1}{2}}n+1,{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {3}{2}},-{\operatorname {tang} t}^{2})$

XX. $\cos nt=F({\tfrac {1}{2}}n,-{\tfrac {1}{2}}n,{\tfrac {1}{2}},{\sin t}^{2})$

XXI. $\cos nt=\cos t.F({\tfrac {1}{2}}n+{\tfrac {1}{2}},-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}},{\sin t}^{2})$

XXII. $\cos nt={\cos t}^{n}F(-{\tfrac {1}{2}}n,-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}},-{\operatorname {tang} t}^{2})$

XXIII. $\cos nt={\cos t}^{-n}F({\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}}n,{\tfrac {1}{2}},-{\operatorname {tang} t}^{2})$