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.

Read more about this topic:  Mean Value Theorem

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