Cauchy-Riemann condition

[chwala]

Homework Statement

Verify $f(z)= z^3-5iz+√7$ satisfies cauchy riemann equations.

Homework Equations

The Attempt at a Solution

seeking alternative method
$f(z)= (x^3+5y-3xy^2+√7)+ (3x^2y-y^3-5x)i$
$∂u/∂x = 3x^2-3y^2 = ∂v/∂y$
$∂v/∂x=6xy-5= -∂u/dy$
hence satisfies.

 

[jambaugh]

Your attempt appears correct. What was your question?

 

[Mark44]

Homework Statement

Verify $f(z)= z^3-5iz+√7$ satisfies cauchy riemann equations.

Homework Equations

The Attempt at a Solution

seeking alternative method

Why? Is there some reason that the work below isn't satisfactory?

$f(z)= (x^3+5y-3xy^2+√7)+ (3x^2y-y^3-5x)i$
$∂u/∂x = 3x^2-3y^2 = ∂v/∂y$
$∂v/∂x=6xy-5= -∂u/dy$
hence satisfies.

 

[Dick]

seeking alternative method

If you think about expressing $f=u+iv$ in terms of $z$ and $\bar{z}$ instead of $x$ and $y$, it's easy to show that the CR equations say that $\frac{\partial{f}}{\partial{\bar{z}}}=0$. So any function that depends only on $z$ will satisfy CR.

 

[FactChecker]

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.

 

[chwala]

Why? Is there some reason that the work below isn't satisfactory?

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?

 

[chwala]

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.

thats exactly what i did. i substituted for $z=x+iy$

 

[chwala]

Your attempt appears correct. What was your question?

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]

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:
$$\Delta u = \Delta v = 0$$
but while this is a necessary condition it is not sufficient. It may be satisfied for functions failing to satisfy CR.
[edit: oops for the typo... now corrected.]

 

[chwala]

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:
$$\Delt u = \Delta v = 0$$
but while this is a necessary condition it is not sufficient. It may be satisfied for functions failing to satisfy CR.

...directly verifiable...thank you.

 

[FactChecker]

thats exactly what i did. i substituted for $z=x+iy$

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]

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?

kindly note pde means partial differential equations...just for clarity