KitKattt

Complex Analysis_ Holomorphic Functions

Let f be a holomorphic function on some domain Ω. Let z0 ∈ Ω and define g(z) = (f(z)−f(z0) z−z0 , z 6= z0 , f 0 (z0), z = z0 . Show that one can express g(z) as g(z) = Z 1 0 (f 0 ◦ w)(t)dt , where w : [0, 1] → C : t 7→ (z − z0)t + z0 is the line segment from z0 to z. By considering f 00 , prove that g(z) is holomorphic in Ω. Hence, deduce that g(z) = ( sin z z , z 6= 0 , 1 , z = 0 , (0.1) is an entire function. (The singularity of the function sin z z at z = 0 is referred to as a removable singularity.)
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