DEF: f f f : Ω → C :Omega->CC : Ω → C komplex differenzierbar in z 0 ∈ Ω : ⇔ z_0 in Omega:<=> z 0 ∈ Ω :⇔ f ´ ( z 0 ) : = lim z → z 0 f ( z ) − f ( z 0 ) z − z 0 f´(z_0) := lim_(z -> z_0) (f(z) - f(z_0)) / (z - z_0) f ´ ( z 0 ) := lim z → z 0 z − z 0 f ( z ) − f ( z 0 ) Ω ⊂ C Omega subset CC Ω ⊂ C DEF: f f f auf Ω : ⇔ Omega :<=> Ω :⇔ f ∈ O ( Ω ) : ⇔ f in cal(O)(Omega):<=> f ∈ O ( Ω ) :⇔ f f f z ∈ Ω z in Omega z ∈ Ω
SATZ: : f f f z 0 z_0 z 0 ⇒ => ⇒ f f f z 0 z_0 z 0 Beweis: lim z → z 0 f ( z ) = f ( z 0 ) lim_(z -> z_0) f(z) = f(z_0) lim z → z 0 f ( z ) = f ( z 0 ) ⇔ lim z → z 0 ( f ( z ) − f ( z 0 ) ) = 0 <=> lim_(z -> z_0) (f(z) - f(z_0)) = 0 ⇔ lim z → z 0 ( f ( z ) − f ( z 0 ) ) = 0 lim z → z 0 ( f ( z ) − f ( z 0 ) ) lim_(z -> z_0) (f(z) - f(z_0)) lim z → z 0 ( f ( z ) − f ( z 0 ) ) = lim z → z 0 ( f ( z ) − f ( z 0 ) z − z 0 ⋅ ( z − z 0 ) ) = lim_(z -> z_0) ( (f(z) - f(z_0)) / (z - z_0) dot (z - z_0) ) = lim z → z 0 ( z − z 0 f ( z ) − f ( z 0 ) ⋅ ( z − z 0 ) ) = ( lim z → z 0 f ( z ) − f ( z 0 ) z − z 0 ) ⋅ ( lim z → z 0 ( z − z 0 ) ) = ( lim_(z -> z_0) (f(z) - f(z_0)) / (z - z_0) ) dot ( lim_(z -> z_0) (z - z_0) ) = ( lim z → z 0 z − z 0 f ( z ) − f ( z 0 ) ) ⋅ ( lim z → z 0 ( z − z 0 ) ) = f ´ ( z 0 ) ⋅ 0 = f´(z_0) dot 0 = f ´ ( z 0 ) ⋅ 0 = 0 = 0 = 0
SATZ: ∂ x f ( g ( x ) = ∂ g f ( g ( x ) ) ⋅ ∂ x g ( x ) diff_x f(g(x)=diff_g f(g(x)) dot diff_x g(x) ∂ x f ( g ( x ) = ∂ g f ( g ( x )) ⋅ ∂ x g ( x ) ⇔ ∂ f ( g ( x ) ) ∂ x = ∂ f ( g ( x ) ) ∂ g ( x ) ⋅ ∂ g ( x ) ∂ x <=> (diff f(g(x)))/(diff x)=(diff f(g(x)))/(diff g(x)) dot (diff g(x))/(diff x) ⇔ ∂ x ∂ f ( g ( x )) = ∂ g ( x ) ∂ f ( g ( x )) ⋅ ∂ x ∂ g ( x ) ⇔ ( f ( g ( x ) ) ) ‘ = f ‘ ( g ( x ) ) ⋅ g ‘ ( x ) <=> (f(g(x)))`=f`(g(x))dot g`(x) ⇔ ( f ( g ( x ))) ‘ = f ‘ ( g ( x )) ⋅ g ‘ ( x ) ⇔ ( f ○ g ) ↓ x ( x ) = f ↓ g ( g ( x ) ) ⋅ g ↓ x ( x ) <=>(f circle g) ^(arrow.b x) (x) = f^(arrow.b g)(g(x)) dot g^(arrow.b x) (x) ⇔ ( f ○ g ) ↓ x ( x ) = f ↓ g ( g ( x )) ⋅ g ↓ x ( x )