Could you help with solving this complex integral: $$\int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz$$ where $C$ is $|z|=5$. I am expecting that the Residue Theorem will be needed. The answer should be $i\pi$.
2026-03-28 00:47:46.1774658866
Complex integral and Laurent series
132 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in INTEGRATION
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- How to find the unit tangent vector of a curve in R^3
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- Proving smoothness for a sequence of functions.
- Random variables in integrals, how to analyze?
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Which type of Riemann Sum is the most accurate?
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 LAURENT-SERIES
- Find Laurent series of rational function $f(z)={1 \over (z+1)^2(z+2)}$
- How do I show with Laurent Series Expansion that $1/z$ has a simple pole for $z=z_0=0$?
- Order of Poles of $1/\cos(1/z)$
- Classification of singularities of $\sin\left( \frac{1}{\sin(\frac{1}{z})}\right)$
- Laurent expansion and singularities of $\frac{1-\cos(z)}{e^{2iz}-1}$
- Laurent Series problems
- Laurent series VS Fourier series.
- Laurent series and radius of convergence of $f(z)=\frac{1}{(1-\cosh z)^2}$
- Show that a localization a power series ring $R[[x]]$ by $S$ can be written a certain way.
- Laurent series of complex function
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?
Hint:
$$z^3e^{-1/z^2} = z^3 - \frac{1}{1!} z + \frac{1}{2!} \frac{1}{z} - \frac{1}{3!} \frac{1}{z^3} + \frac{1}{4!} \frac{1}{z^5} - \cdots$$
What happens when you take your contour integral of that sum? Which terms have non-zero residue?