I am wondering if there is an anti-derivative of the complex absolute value and if it exists, how to find it.
I am looking for
$ \int |z|^2 dz = ? $
Any help is appreciated!
2026-04-08 11:02:06.1775646126
Finding the anti-derivative of complex absolute value
80 Views Asked by user76844 https://math.techqa.club/user/user76844/detail At
1
There are 1 best solutions below
Related Questions in COMPLEX-ANALYSIS
- Minkowski functional of balanced domain with smooth boundary
- limit points at infinity
- conformal mapping and rational function
- orientation of circle in complex plane
- If $u+v = \frac{2 \sin 2x}{e^{2y}+e^{-2y}-2 \cos 2x}$ then find corresponding analytical function $f(z)=u+iv$
- Is there a trigonometric identity that implies the Riemann Hypothesis?
- order of zero of modular form from it's expansion at infinity
- How to get to $\frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz =n_0-n_p$ from Cauchy's residue theorem?
- If $g(z)$ is analytic function, and $g(z)=O(|z|)$ and g(z) is never zero then show that g(z) is constant.
- Radius of convergence of Taylor series of a function of real variable
Related Questions in INDEFINITE-INTEGRALS
- Closed form of integration
- How to find $\int \sqrt{x^8 + 2 + x^{-8}} \,\mathrm{d}x$?
- Find the integral $\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\,dx.$
- Integrate $\int \frac {x^4}{\sqrt {x^2-9}} \,dx$
- Integral of $\frac{1}{2x}$.
- Contradictory results of the integral of an odd function
- Integrate $\int \frac{x+2}{(x^2+3x+3) \sqrt{x+1}} dx$
- Evaluation of Integral $\int \frac{x^2+1}{\sqrt{x^3+3}}dx$
- Integral of a Polynomial in Square Root
- Using a substitution of a square of a trigonometric function.
Related Questions in ABSOLUTE-VALUE
- To find the Modulus of a complex number
- What does $|a| = |b|$ is equal to?
- Symmetric polynomial written in elementary polynomials
- If $|ax^2+bx+c|\le \frac12$ for all $|x|\le1$, then $|ax^2+bx+c|\le x^2-\frac12$ for all $|x|\ge1$
- Proving that a double integral converges
- Equation system
- If $\sqrt{9−8\cos 40^{\circ}} = a +b\sec 40^{\circ}$, then what is $|a+b|$?
- Proving that inequalities $\|a\|_{\infty} \leq \|a\|_2 \leq \sqrt{n} \|a\|_{\infty}$ are true and sharp.
- Find a number $M$, such that $|x^3-4x^2+x+1| < M$ for all $1<x<3$
- Absolute Value of a Complex Number Inequality
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
No. One the complex plane, only analytic functions have derivatives or anti-derivatives. And neither $|z|$ nor $|z|^2$ is analytic anywhere. These functions are real-valued everywhere, and the only complex-analytic functions whose images are not 2-dimensional are constants.
Integration in $\Bbb C$ is along curves. So to have an integral, you must define the curve it is along. The only exception is when the funtion you are analyzing is conservative, so the integral along any closed curve is always $0$. In that case, it doesn't matter which path you take between two points - the value of the integral is dependent on only the end points. However in $\Bbb C$, the condition for being conservative happens to coincide with the condition for being analytic. Thus you cannot use integration to define an anti-derivative unless the integrand is analytic.