How can I use the residue theorem to evaluate the following integral $$ \int_{-\infty}^{\infty} \frac{|x|^{1/2}}{(x-4i)(x+2i)}dx $$ I think my trouble is in handling the absolute value. Should something special be done about it and is there a general way to do that?
2026-04-06 08:02:10.1775462530
Solving integral involving absolute value by residue theorem
222 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated 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
Related Questions in RESIDUE-CALCULUS
- 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?
- contour integral involving the Gamma function
- The Cauchy transform of Marchenko-Pastur law
- Contour Integration with $\sec{(\sqrt{1-x^2})}$
- calculate $\int_{-\infty}^\infty\frac{e^{ix} \, dx}{x^3-3ix^2+2x+2i}$
- Integral $\int_{-\infty}^{\infty} \frac{ \exp\left( i a e^{u}\right) }{ e^{b \cosh(u)} - 1 } du$
- Solve the improper integral with techniques of complex analysis
- Compute the integral with use of complex analysis techniques
- $\int\limits_{-\infty}^\infty \frac{1}{e^{x^{2}}+1}dx$
- Residue Theorem: Inside vs. Outside
Related Questions in COMPLEX-INTEGRATION
- Contour integration with absolute value
- then the value of $ \frac{1-\vert a \vert^2}{\pi} \int_{\gamma} \frac{\vert dz \vert}{\vert z+a \vert^2} $.
- Checking that a function is in $L^p(\mathbb{C})$
- Calculate integral $\int_{0}^{2\pi} \frac{dx}{a^2\sin^2x+b^2\cos^2x}$
- Complex integral of $\cfrac{e^{2z}}{z^4}$
- Have I solved this complex gaussian integral correctly?
- Evaluate the integral $ I=\frac{1}{2\pi i}\int_{\vert z \vert =R}(z-3)\sin \left(\frac{1}{z+2}\right)dz$,
- Integrating using real parts
- Complex integral(s)of Hyperbolic functions for different contours
- Are the Poles inside the contour?
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?