Mean Value Theorem - Generalization in Complex Analysis

Generalization in Complex Analysis

As noted above, the theorem does not hold for differentiable complex-valued functions. Instead, a generalization of the theorem is sated such:

Topics in calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential calculus Derivative Second derivative Third derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities Power rule Product rule Quotient rule Sum rule Chain rule
Integral calculus Lists of integrals Improper integral Multiple integral Integration by parts disks cylindrical shells substitution trigonometric substitution partial fractions changing order
Vector calculus Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem
Multivariable calculus Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

Let be a holomorphic function on the open convex set, and let and be distinct points in . Then there exist points u,v on (Linear line from to ) such that

Where Re is the Real part and Im is the Imaginary part of a complex-valued function.

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