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Redirected from Cauchy-Riemann). In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic

In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations.

Cauchy–Riemann equations. In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. 1 a ) ∂ u ∂ x ∂ v ∂ y ( 1 b ) ∂ u ∂ y − ∂ v ∂ x {displaystyle {begin{aligned}(1a)qquad &{frac {partial u}{partial x}} {frac {partial v}{partial y}}\(1b)qquad &{frac {partial u}{partial y}} -{frac {.

The Cauchy-Riemann equations (3-16) are not satisfied at any point, so we conclude that. is nowhere differentiable. Show that the function defined by. is not differentiable at the point even though the Cauchy-Riemann equations (3-16) are satisfied at the point. We must use limits to calculate the partial derivatives at.,,,. Thus, we can see that. Hence the Cauchy-Riemann equations (3-16) hold at the point Example . reiterates that the mere satisfaction of the Cauchy-Riemann equations is not sufficient to guarantee the differentiability. The following theorem, however, gives conditions that guarantee the differentiability of at, so that which we. can use Equation (3-14) or (3-15) to compute. They are referred to as the Cauchy-Riemann conditions for differentiability. Theorem . (Cauchy-Riemann conditions for differentiability).