All Exercises (with Solutions) containing the “integral_of_holomorphic_function” or “residual_theorem” tag:

General Solution

Useful Knowledge

\int_{- \infty}^{\infty} e^{- x^{2}} = π

Cheat Sheet

MMfE - Cauchy Residue Theorem
Cauchy Residue Theorem

Mathematical_Methods_for_Engineering Theorem
$\int_{γ} f (z) d z = 2 πi i \sum Res {f (z), p_{i}}$
Where:

$f (z)$ is holomorphic

$γ$ is a simple closed positively oriented contour

$p_{i}$ are the poles of $f (z)$

NOTE:

If the curve $γ$ is defined in the ANTI-CLOCK wise $↺$ sense then its $+ 2 πi$ .

If the curve $γ$ is defined in the CLOCK wise $↻$ sense then its $- 2 πi$ .

Residues:

$R es {f (z), p_{i}}$ is calculated as follows:

If $p_{i}$ is outside the closed contour $γ$ :

$R es {f (z), p_{i}} = 0$

If $p_{i}$ is inside $γ$ , and is a simple pole (multiplicity = 1):

$R es {f (z), p_{i}} = z \to p_{i} lim (z - p_{i}) f (z)$

If $p_{i}$ is inside $γ$ , and is a complex pole of multiplicity = $k$ :

$R es {f (z), p_{i}} = \frac{1}{( k - 1 )!} z \to p_{i} lim \frac{d ^{k - 1}}{d z ^{k - 1}} {(z - p_{i})^{k} f (z)}$
- [[MMfE - Holomorphic or Analytical Functions|What's an Holomorphic Function?]] - [[MMfE - Simple Closed Positively Oriented Contour|What's a Simple Closed Positively Oriented Contour?]] - [[MMfE - Poles of a Function|What are the Poles of a Function?]] - [[MMfE - Multiplicity|What's the Multiplicity?]]

Original File:

Pasted image 20220604185021.png IL LIBRO HA SBAGLIATO, C’È UN ERRORE NELL’ULTIMA FORMA! Integrals of holomorphic functions_220518_145829.pdf cauchy residue theorem_220518_160518.pdf
Link to original

MMfE - Jordan's Lemma
Jordan’s Lemma

Mathematical_Methods_for_Engineering Theorem

But instead of integrating on a closed curve $γ$ we have:
$f (w) = \frac{1}{2 πi} \int_{- \infty}^{\infty} \frac{f ( z )}{z - w} d z$

Original File:

Differential Equations using Laplace Transform_220512_145931.pdf
Link to original

~Ex.: 2020_06_10 Point 1.

Link to original

~Ex.: 2020_06_10 Point 2.

Link to original

~Ex.: 2020_06_10 Point 3.

Link to original

~Ex.: 2020_06_10 Point 4

Link to original

~Ex.: 2020_06_24 Point 1.

Link to original

~Ex.: 2020_06_24 Point 2.

Link to original

~Ex.: 2020_06_24 Point 3.

Link to original

~Ex.: 2020_06_24 Point 4.

Link to original

~Ex. 2020_07_15 Point 1.

Link to original

~Ex. 2020_07_15 Point 2.

Link to original

~Ex. 2020_07_15 Point 3.

Link to original

~Ex. 2020_07_15 Point 4.

Link to original

Transclude of MMfE-~-exam_2021_02_17---with-solutions#ex-2021_02_17-point-1

Transclude of MMfE-~-exam_2021_02_17---with-solutions#ex-2021_02_17-point-2

Transclude of MMfE-~-exam_2021_02_17---with-solutions#ex-2021_02_17-point-3

Transclude of MMfE-~-exam_2021_02_17---with-solutions#ex-2021_02_17-point-4

~Ex.: 2021_06_25 Point 1.

Link to original

~Ex.: 2021_06_25 Point 2.

Link to original

~Ex.: 2021_06_25 Point 3.

Link to original

~Ex.: 2021_06_25 Point 4.

Link to original

MMfE ~ exam_2021_09_10 - with solutions
Contents: #Mathematical_Methods_for_Engineering #exercises_with_solutions

Exercise 1 1.1.sketch_of_complex_curve 1.2.integral_of_holomorphic_function residual_theorem

Exercise 2 2.1.region_of_holomorphicity type_of_isolated_singularities 2.2.coefficients_of_Laurent_series region_of_convergence

Exercise 3 3.1.is_Laplace_transformable Laplace_transform 3.2.sketch_of_complex_curve complex_limit initial_value_theorem

Exercise 4 4.1.solve_differential_equation_using_Laplace_transform

Questions

~Ex.: 2021_09_10 Point 1.

~Ex.: 2021_09_10 Point 2.

~Ex.: 2021_09_10 Point 3.

~Ex.: 2021_09_10 Point 4.

Original Files

Link to original