# Find v such that f(z) = u+iv is analytic

• Vitani11
In summary, the problem is to find v in terms of u such that f(z) = u+iv is analytic. This can be done by using the Cauchy-Riemann equations and the Laplace equation to manipulate the partial derivatives of u and v. The approach is to set up an integral for v and add a function of x to account for the vanishing term when differentiating with respect to y. The given problem statement is ambiguous and may require additional information to solve.
Vitani11

## Homework Statement

Find v such that f(z) = u+iv is analytic.

## Homework Equations

du/dx = dv/dy and dv/dx=-du/dy

## The Attempt at a Solution

I'm not sure what I'm supposed to do. I think I need to find U in order to find V because if a function is analytic it satisfies the Cauchy Riemann equations. I tried to play around with cauchy riemann equations to get dv in terms of everything else but that's not helping. Can I also use the Laplace equation to solve this? This is so general that I know it's probably simple, but at the same time it makes it hard to understand what my answer is supposed to look like. What should my approach be?

You can find v as function of u.
It is a bit confusing to use the same symbols for the complex components of the argument (first equation) and the complex components of the function ("relevant equations").

Vitani11
Suppose you are given $u:=u(x,y)$ and you know $f$ is differentiable so it satisfies C-R, in particular,$\frac{du}{dx} = \frac{dv}{dy}$.
Therefore $v = \int u_x dy + g(x)$, where $u_x$ is the partial derivative w.r.t $x$. You add a function of $x$ because when you differentiate w.r.t $y$, the $g(x)$ vanishes.

Last edited:
Vitani11 and FactChecker
That's not how I would interpret the u and v given in the first post.

@Vitani11: Is that the full and exact problem statement? "Find v such that f(z) = u+iv is analytic." Nothing else given?

Vitani11
The problem has to start from somewhere. We must either have $u$ or $v$. I agree that the problem is ambiguous.

Vitani11
That is 100% all that was given, promise. Anyway thank you I now think I can solve this.

## 1. What does it mean for a function to be "analytic"?

An analytic function is one that can be represented by a convergent power series in a given domain. This means that the function is continuously differentiable and can be approximated by polynomials with an infinite radius of convergence.

## 2. How do I find v for a given function f(z)?

To find v for a given function f(z), you can use the Cauchy-Riemann equations. These equations relate the real and imaginary parts of a complex function and can help you solve for v.

## 3. Can I find v if I only know the real part of the function f(z)?

Yes, you can still find v if you only know the real part of the function f(z). You will need to use the Cauchy-Riemann equations to solve for v using the real part of the function.

## 4. Are there any shortcuts for finding v in an analytic function?

There are some shortcuts for finding v in an analytic function, such as using the properties of logarithms and exponentials. However, these shortcuts may not always be applicable and it is important to understand the Cauchy-Riemann equations for a more general approach.

## 5. Can I use complex analysis to find v for any function?

No, complex analysis can only be used to find v for analytic functions. Non-analytic functions may not have a well-defined v, and therefore, complex analysis cannot be used to find it.

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