DNU365; Date:3/7/20
Augustin Cauchy 's and Bernhard Riemann brilliancy towards resolving complex derivative issues by bringing their equations known as C-R equations.
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As per calculus theory which one have read during his life would have found that for real numbers it's very easy to find differentiability for a point. We just have to chase a point from left hand side to right hand side. As LHD=RHD we say it's differentiable. But just think a moment when the real function is replaced by complex function was it that easy...
No, not at all it just bashed heads of all brillianties during that time and no one would find solution to show differentiability of complex function.
Reason: As in real function we have only two paths for a point from left hand side to right hand side , but in a complex function there are undefined paths ...we could even say there are infinite paths and for that it was just impossible to show the all the paths from LHD to IHD (Infinite hand derivative) equal. So this was a point for concern ..how the problem resolve.
Mathematicians known by name of cauchy and Rieman have tried to resolve the issue by providing one solution regarding this.They said that if partial derivative of complex function be equal i.e for f(z)=U(x,y)+iV(x,y) if
U(x)=V(y) and U(y)=-V(x) by this equation he amazed everyone as issues on differentiability was resolved becouse all complex functions were showing results through this. But there some problem arises and some functions were differentiable but not showing results so what Cauchy &Riemann did then:
1) Cauchy &Riemann than said if function is differentiable than it would satisfy their equation i.e C.R equation and if C.R equation not satisfied than function is not differentiable and referred it as necessary condition but it was not 100 percent solution towards issues.
2) Later Cauchy &Riemann with an view to resolved it 100 percent came with sufficient condition and said for a function to be differentiable it should follow C.R equation and also it's component would exist and be continuous throughout path ....
Cauchy&Rieman this contribution cannot be forgoten ever which lead a very significant role in bringing complex function calculus ....
