f : ( − r , r ) → R {\displaystyle f:(-r,r)\rightarrow \mathbb {R} }
cosh x = 1 2 ( f 8 x ) + f ( − x ) ⏟ g r a d e F k t . + 1 2 ( f ( x ) − f ( x ) ) ⏟ u n g r a d e F k t . {\displaystyle \cosh x=\underbrace {{\frac {1}{2}}(f8x)+f(-x)} _{gradeFkt.}+\underbrace {{\frac {1}{2}}(f(x)-f(x))} _{ungradeFkt.}}
Anmerkung: wofür steht FKt.?
cosh x = 1 2 ( e x + e − x ) {\displaystyle \cosh x={\frac {1}{2}}(e^{x}+e^{-}x)} cosinus hyperbolicus ∑ n = 0 ∞ x 2 n ( 2 n ) ! ( x ∈ R ) {\displaystyle \sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}(x\in \mathbb {R} )}
sinh x = 1 2 ( e x − e − x ) {\displaystyle \sinh x={\frac {1}{2}}(e^{x}-e^{-}x)} sinus hyperbolicus ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! ( x ∈ R ) {\displaystyle \sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}(x\in \mathbb {R} )}
tanh x = sinh x cosh x {\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}} tangens hyperbolicus e 2 x − 1 e 2 x + 1 ( x ∈ R ) {\displaystyle {\frac {e^{2x}-1}{e^{2x}+1}}(x\in \mathbb {R} )}
coth x = cosh x sinh x {\displaystyle \coth x={\frac {\cosh x}{\sinh x}}} cotangens hyperbolicus e 2 x + 1 e 2 x − 1 ( x ≠ 0 ) {\displaystyle {\frac {e^{2x}+1}{e^{2x}-1}}(x\not =0)}
hier fehlt eine Zeichung
( cosh t ) 2 = 1 4 ( e t + e − t ) 2 = 1 4 ( e 2 t + 2 + e − 2 t ) {\displaystyle (\cosh t)^{2}={\frac {1}{4}}(e^{t}+e^{-t})^{2}={\frac {1}{4}}(e^{2t}+2+e^{-2t})}
( s i n h t ) 2 = 1 4 ( e t + e − t ) 2 == 1 4 ( e 2 t − 2 + e − 2 t ) {\displaystyle (sinht)^{2}={\frac {1}{4}}(e^{t}+e^{-t})^{2}=={\frac {1}{4}}(e^{2t}-2+e^{-2t})}
( cosh t ) 2 − ( sinh t ) 2 = 1 {\displaystyle (\cosh t)^{2}-(\sinh t)^{2}=1}
lim x → ∞ coth x = 1 lim x → ∞ tanh x {\displaystyle \lim _{x\rightarrow \infty }\coth x=1\lim _{x\rightarrow \infty }\tanh x}
lim x → − ∞ coth x = − 1 lim x → − ∞ tanh x {\displaystyle \lim _{x\rightarrow -\infty }\coth x=-1\lim _{x\rightarrow -\infty }\tanh x}
hier fehlt etwas
t ∈ R {\displaystyle t\in \mathbb {R} }
e i t := ∑ n = 0 ∞ ( i t ) n n ! = ∑ m = 0 ∞ ( − 1 ) m t 2 m ( 2 m ) ! + i ∑ m = 0 ∞ ( − 1 ) m 2 m + 1 t 2 m + 1 {\displaystyle e^{it}:=\sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}=\sum _{m=0}^{\infty }{\frac {(-1)^{m}t^{2}m}{(2m)!}}+i\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{2m+1}}t^{2m+1}}
i 2 m = ( − 1 ) m , i 2 m + 1 = i 2 m + 1 = i ( − 1 ) m {\displaystyle i^{2m}=(-1)^{m},i^{2m+1}=i^{2m+1}=i(-1)^{m}}
cos x = ∑ {\displaystyle \cos x=\sum _{}^{}}