Why? Is there some reason that the work below isn't satisfactory?chwala said:Homework Statement
Verify ## f(z)= z^3-5iz+√7## satisfies cauchy riemann equations.Homework Equations
The Attempt at a Solution
seeking alternative method
chwala said:## f(z)= (x^3+5y-3xy^2+√7)+ (3x^2y-y^3-5x)i##
##∂u/∂x = 3x^2-3y^2 = ∂v/∂y##
##∂v/∂x=6xy-5= -∂u/dy##
hence satisfies.
chwala said:seeking alternative method
why? i have shown that both sides of the cauchy riemann pde hold. that's a summary of my working, or you want to see the full working?Mark44 said:Why? Is there some reason that the work below isn't satisfactory?
thats exactly what i did. i substituted for ##z=x+iy##FactChecker said:There is a simple way to see it immediately if you know that the sums and products of functions that satisfy CR also satisfy CR. Constants and f(z)=z satisfy CR. Your equation is a combination of sums and products satisfying CR, so it also satisfies CR.
My method is correct, its not an attempt. I was proving or rather to show that the given ##f(z)## satisfies the Cauchy Riemann equations. I am simply asking for alternative ways of solving. regards,jambaugh said:Your attempt appears correct. What was your question?
...directly verifiable...thank you.jambaugh said:Well as the Cauchy Riemann conditions are directly verifiable there's not much else you can do but directly verify them. There are consequential conditions like the harmonic condition:
[tex]\Delt u = \Delta v = 0[/tex]
but while this is a necessary condition it is not sufficient. It may be satisfied for functions failing to satisfy CR.
What you did is not the same as applying general principles. You don't have to do the messy calculations if the principles have been proven. You have to have proven the principle that sums and multiplications of functions satisfying CR will also satisfy CR. Then just state that your function is a combination of sums and multiplications of simple functions satisfying CR and apply the principle.chwala said:thats exactly what i did. i substituted for ##z=x+iy##
kindly note pde means partial differential equations...just for claritychwala said:why? i have shown that both sides of the cauchy riemann pde hold. that's a summary of my working, or you want to see the full working?
The Cauchy-Riemann condition is a set of necessary and sufficient conditions for a complex-valued function to be differentiable at a point in the complex plane. It states that the partial derivatives of the real and imaginary components of the function must satisfy certain relationships at that point.
The Cauchy-Riemann condition is important because it is one of the fundamental conditions for a function to be considered analytic. This means that the function is infinitely differentiable and can be represented by a power series. Analytic functions have many useful properties and are widely used in many areas of mathematics and science.
The Cauchy-Riemann condition is closely related to the concept of holomorphic functions. A function that satisfies the Cauchy-Riemann condition at all points in a given domain is said to be holomorphic in that domain. Holomorphic functions have many important properties, such as being conformal and having a unique analytic continuation.
Yes, the Cauchy-Riemann condition can be extended to higher dimensions. In three dimensions, it takes the form of the Cauchy-Riemann equations, which are a system of four partial differential equations. In general, the Cauchy-Riemann condition can be extended to any number of dimensions, but the equations become increasingly complex.
The Cauchy-Riemann condition has many applications in mathematics and physics. It is used in complex analysis, which has applications in fields such as fluid dynamics, signal processing, and quantum mechanics. The condition also plays a crucial role in the study of conformal mappings, which are important in fields such as cartography and computer graphics.