The residue theorem is effectively a generalization of Cauchy's integral formula. Pr Proof. Because residues rely on the understanding of a host of topics such as the nature of the logarithmic function, integration in the complex plane, and Laurent series, it is recommended that you be familiar with all of these topics before proceeding. This will allow us to compute the integrals in Examples 4.8-4.10 in an easier and less ad hoc manner. The Cauchy residue theorem can be used to compute integrals, by choosing the appropriate contour, looking for poles and computing the associated residues. One is inside the unit circle and one is outside.) I followed the derivation of the residue theorem from the cauchy integral theorem and I think I kinda understand what is going on there. In an upcoming topic we will formulate the Cauchy residue theorem. Theorem 31.4 (Cauchy Residue Theorem). Suppose that C is a closed contour oriented counterclockwise. There will be two things to note here. Walk through homework problems step-by-step from beginning to end. Ref. Let C be a closed curve in U which does not intersect any of the a i. Theorem 45.1. For these, and proofs of theorems such as Fundamental Theorem of Algebra or Louiville’s theorem you never need more than a finite number of arcs and lines (or a circle – which is just a complete arc). We perform the substitution z = e iθ as follows: Apply the substitution to thus transforming them into . REFERENCES: Arfken, G. "Cauchy's Integral Theorem." We recognize that the only pole that contributes to the integral will be the pole at, Next, we use partial fractions. Once we do both of these things, we will have completed the evaluation. Calculation of Complex Integral using residue theorem. wikiHow is where trusted research and expert knowledge come together. When f : U ! If is any piecewise C1-smooth closed curve in U, then Z f(z) dz= 0: 3.3 Cauchy’s residue theorem Theorem (Cauchy’s residue theorem). REFERENCES: Arfken, G. "Cauchy's Integral Theorem." 48-49, 1999. It will turn out that $$A = f_1 (2i)$$ and $$B = f_2(-2i)$$. Theorem 31.4 (Cauchy Residue Theorem). If f(z) is analytic inside and on C except at a ﬁnite number of isolated singularities z 1,z 2,...,z n, then C f(z)dz =2πi n j=1 Res(f;z j). (11) can be resolved through the residues theorem (ref. To create this article, volunteer authors worked to edit and improve it over time. By Cauchy’s theorem, this is not too hard to see. Unlimited random practice problems and answers with built-in Step-by-step solutions. Using residue theorem to compute an integral. Because residues rely on the understanding of a host of topics such as the nature of the logarithmic function, integration in the complex plane, and Laurent series, it is recommended that you be familiar with all of these topics before proceeding. Let f (z) be analytic in a region R, except for a singular point at z = a, as shown in Fig. (c)Thefunctionlog αisanalyticonC\R,anditsderivativeisgivenbylog α(z)=1/z. Remember that out of four fractions in the expansion, only the term, Notice that this residue is imaginary - it must, if it is to cancel out the. When f: U!Xis holomorphic, i.e., there are no points in Uat which fis not complex di erentiable, and in Uis a simple closed curve, we select any z 0 2Un. The Cauchy residue theorem can be used to compute integrals, by choosing the appropriate contour, looking for poles and computing the associated residues. We can factor the denominator: f(z) = 1 ia(z a)(z 1=a): The poles are at a;1=a. Suppose that C is a closed contour oriented counterclockwise. Then the integral in Eq. However you do it, you get, for any integer k , I C0 (z − z0)k dz = (0 if k 6= −1 i2π if k = −1. Find more Mathematics widgets in Wolfram|Alpha. Cauchy integral and residue theorem [closed] Ask Question Asked 1 year, 2 months ago. From this theorem, we can define the residue and how the residues of a function relate to the contour integral around the singularities. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. 2.But what if the function is not analytic? On the circle, write z = z 0 +reiθ. Proof. For these, and proofs of theorems such as Fundamental Theorem of Algebra or Louiville’s theorem you never need more than a finite number of arcs and lines (or a circle – which is just a complete arc). Cauchy residue theorem. (Residue theorem) Suppose U is a simply connected … Chapter & Page: 17–2 Residue Theory before. PDF | On May 7, 2017, Paolo Vanini published Complex Analysis II Residue Theorem | Find, read and cite all the research you need on ResearchGate Cauchy residue theorem. integral for any contour in the complex plane In an upcoming topic we will formulate the Cauchy residue theorem. Definition. The Cauchy Residue Theorem Before we develop integration theory for general functions, we observe the following useful fact. 0) = 1 2ˇi Z. Cauchy’s residue theorem — along with its immediate consequences, the ar- gument principle and Rouch ´ e’s theorem — are important results for reasoning We note that the integrant in Eq. You can compute it using the Cauchy integral theorem, the Cauchy integral formulas, or even (as you did way back in exercise 14.14 on page 14–17) by direct computation after parameterizing C0. A generalization of Cauchy’s theorem is the following residue theorem: Corollary 1.5 (The residue theorem) f ∈ Cω(D \{zi}n i=1), D open containing {zi} with boundary δD = γ. Then $\int_{C} f(z) \ dz = 2\pi i \sum \text{ residues of } f \text{ inside } C$ Proof. Proposition 1.1. A theorem in complex analysis is that every function with an isolated singularity has a Laurent series that converges in an annulus around the singularity. The integral in Eq. Well, it means you have rigorously proved a version that will cope with the main applications of the theorem: Cauchy’s residue theorem to evaluation of improper real integrals. gives, If the contour encloses multiple poles, then the 1. To create this article, volunteer authors worked to edit and improve it over time. Proof. Thus for a curve such as C 1 in the figure §6.3 in Mathematical Methods for Physicists, 3rd ed. wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. Proof. §33 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. Important note. the contour. 0. This will allow us to compute the integrals in Examples 5.3.3-5.3.5 in an easier and less ad hoc manner. In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Suppose that f(z) is analytic inside and on a simply closed contour C oriented counterclockwise. 5.3 Residue Theorem. 6. proof of Cauchy's theorem for circuits homologous to 0. Suppose C is a positively oriented, simple closed contour. Cauchy integral and residue theorem [closed] Ask Question Asked 1 year, 2 months ago. However, only one of them lies within the contour - the other lies outside and will not contribute to the integral. Proof. 2 CHAPTER 3. Then ∫ C f ⁢ (z) ⁢ z = 2 ⁢ π ⁢ i ⁢ ∑ i = 1 m η ⁢ (C, a i) ⁢ Res ⁡ (f; a i), where. Seine Bedeutung liegt nicht nur in den weitreichenden Folgen innerhalb der Funktionentheorie, sondern auch in der Berechnung von Integralen über reelle Funktionen. We use the Residue Theorem to compute integrals of complex functions around closed contours. This document is part of the ellipticpackage (Hankin 2006). See more examples in By using our site, you agree to our. It is easy to apply the Cauchy integral formula to both terms. Also suppose is a simple closed curve in that doesn’t go through any of the singularities of and is oriented counterclockwise. 2πi C f(ζ) (ζ −z)n+1 dζ, n =1,2,3,.... For the purposes of computations, it is usually more convenient to write the General Version of the Cauchy Integral Formula as follows. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. https://mathworld.wolfram.com/ResidueTheorem.html, Using Zeta This amazing theorem therefore says that the value of a contour the first and last terms vanish, so we have, where is the complex 2. Suppose that f(z) has an isolated singularity at z0 and f(z) = X∞ k=−∞ ak(z − z0)k is its Laurent expansion in a deleted neighbourhood of z0. An analytic function whose Laurent series is given by(1)can be integrated term by term using a closed contour encircling ,(2)(3)The Cauchy integral theorem requires thatthe first and last terms vanish, so we have(4)where is the complex residue. consider supporting our work with a contribution to wikiHow, We see that the integral around the contour, The Cauchy principal value is used to assign a value to integrals that would otherwise be undefined. The Residue Theorem has Cauchy’s Integral formula also as special case. This document is part of the ellipticpackage (Hankin 2006). Then ∫ C f ⁢ (z) ⁢ z = 2 ⁢ π ⁢ i ⁢ ∑ i = 1 m η ⁢ (C, a i) ⁢ Res ⁡ (f; a i), where. Let U ⊂ ℂ be a simply connected domain, and suppose f is a complex valued function which is defined and analytic on all but finitely many points a 1, …, a m of U. 1 $\begingroup$ Closed. 9 De nite integrals using the residue theorem 9.1 Introduction In this topic we’ll use the residue theorem to compute some real de nite integrals. Er besagt, dass das Kurvenintegral … depends only on the properties of a few very special points inside % of people told us that this article helped them. Cauchy’s residue theorem — along with its immediate consequences, the ar- gument principle and Rouch ´ e’s theorem — are important results for reasoning Suppose $$f(z)$$ is analytic in the region $$A$$ except for a set of isolated singularities. 137-145]. First, we will find the residues of the integral on the left. Keywords: Residue theorem, Cauchy formula, Cauchy’s integral formula, contour integration, complex integration, Cauchy’s theorem. Let U ⊂ ℂ be a simply connected domain, and suppose f is a complex valued function which is defined and analytic on all but finitely many points a 1, …, a m of U. Fourier transforms. It generalizes the Cauchy integral theorem and Cauchy's integral formula. (11) can be resolved through the residues theorem (ref. (11) for the forward-traveling wave containing i (ξ x − ω t) in the exponential function. Question on evaluating $\int_{C}\frac{e^{iz}}{z(z-\pi)}dz$ without the residue theorem. So we will not need to generalize contour integrals to “improper contour integrals”. the contour. 0inside C: f(z. math; Complex Variables, by Andrew Incognito ; 5.2 Cauchy’s Theorem; We compute integrals of complex functions around closed curves. Der Residuensatz ist ein wichtiger Satz der Funktionentheorie, eines Teilgebietes der Mathematik. Viewed 315 times -2. 3.We will avoid situations where the function “blows up” (goes to inﬁnity) on the contour. Active 1 year, 2 months ago. 2.But what if the function is not analytic? In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. Only the simplest version of this theorem is used in this book, where only so-called first-order poles are encountered. (7.2) is i rn−1 Z 2π 0 dθei(1−n)θ, (7.4) which evidently integrates to zero if n 6= 1, but is 2 πi if n = 1. Thanks to all authors for creating a page that has been read 14,716 times. A contour is called closed if its initial and terminal points coincide. The integral in Eq. The classic example would be the integral of. of Complex Variables. Here are classical examples, before I show applications to kernel methods. Then for any z. In general, we use the formula below, where, We can also use series to find the residue. where is the set of poles contained inside Krantz, S. G. "The Residue Theorem." 4 CAUCHY’S INTEGRAL FORMULA 7 4.3.3 The triangle inequality for integrals We discussed the triangle inequality in the Topic 1 notes. 1 2πi Z γ f(z) dz = Xn i=1 Res(f,zi) . Proof: By Cauchy’s theorem we may take C to be a circle centered on z 0. 1. The residue theorem. Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker. 1. [1], p. 580) applied to a semicircular contour C in the complex wavenumber ξ domain. This question is off-topic. This article has been viewed 14,716 times. Method of Residues. We use cookies to make wikiHow great. Cauchy's integral formula helps you to determine the value of a function at a point inside a simple closed curve, if the function is analytic at all points inside and on the curve. Clearly, this is impractical. Boston, MA: Birkhäuser, pp. Let C be a closed curve in U which does not intersect any of the a i. can be integrated term by term using a closed contour encircling , The Cauchy integral theorem requires that Theorem 45.1. All possible errors are my faults. It generalizes the Cauchy integral theorem and Cauchy's integral formula. Knopp, K. "The Residue Theorem." This will allow us to compute the integrals in Examples 5.3.3-5.3.5 in an easier and less ad hoc manner. The residue theorem is effectively a generalization of Cauchy's integral formula. The residue theorem is effectively a generalization of Cauchy's integral formula. Theorem $$\PageIndex{1}$$ Cauchy's Residue Theorem. This question is off-topic. It is not currently accepting answers. §4.4.2 in Handbook Important note. I thought about if it's possible to derive the cauchy integral formula from the residue theorem since I read somewhere that the integral formula is just a special case of the residue theorem. The residue theorem then gives the solution of 9) as where Σ r is the sum of the residues of R 2 (z) at those singularities of R 2 (z) that lie inside C. Details. (11) has two poles, corresponding to the wavenumbers − ξ 0 and + ξ 0. series is given by. Viewed 315 times -2. Also suppose $$C$$ is a simple closed curve in $$A$$ that doesn’t go through any of the singularities of $$f$$ and is oriented counterclockwise. This amazing theorem therefore says that the value of a contour integral for any contour in the complex plane depends only on the properties of a few very special points inside the contour. Consider a second circle C R0(a) centered in aand contained in and the cycle made of the piecewise di erentiable green, red and black arcs shown in Figure 1. In an upcoming topic we will formulate the Cauchy residue theorem. wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. QED. The classical Cauchy-Da venport theorem, which w e are going to state now, is the ﬁrst theorem in additive group theory (see). It will turn out that $$A = f_1 (2i)$$ and $$B = f_2(-2i)$$. Suppose that D is a domain and that f(z) is analytic in D with f (z) continuous. 1.The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. We are now in the position to derive the residue theorem. Because residues rely on the understanding of a host of topics such as the nature of the logarithmic function, integration in the complex plane, and Laurent series, it is recommended that you be familiar with all of these topics before proceeding. If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. The Cauchy Residue Theorem Before we develop integration theory for general functions, we observe the following useful fact. So Cauchy-Goursat theorem is the most important theorem in complex analysis, from which all the other results on integration and differentiation follow. In complex analysis, residue theory is a powerful set of tools to evaluate contour integrals. In general, we can apply this to any integral of the form below - rational, trigonometric functions. We note that the integrant in Eq. An analytic function whose Laurent 1 Residue theorem problems We will solve several problems using the following theorem: Theorem. Explore anything with the first computational knowledge engine. Keywords: Residue theorem, Cauchy formula, Cauchy’s integral formula, contour integration, complex integration, Cauchy’s theorem. The values of the contour (Residue theorem) Suppose U is a simply connected … (11) has two poles, corresponding to the wavenumbers − ξ 0 and + ξ 0.We will resolve Eq. We see that our pole is order 17. 1. Suppose C is a positively oriented, simple closed contour. Hints help you try the next step on your own. [1] , p. 580) applied to a semicircular contour C in the complex wavenumber ξ domain. The following result, Cauchy’s residue theorem, follows from our previous work on integrals. Das Cauchy’sche Fundamentaltheorem (nach Augustin-Louis Cauchy) besagt, dass der Spannungsvektor T (n), ein Vektor mit der Dimension Kraft pro Fläche, eine lineare Abbildung der Einheitsnormale n der Fläche ist, auf der die Kraft wirkt, siehe Abb. All possible errors are my faults. Theorem Cauchy's Residue Theorem Suppose is analytic in the region except for a set of isolated singularities. 1 $\begingroup$ Closed. We apply the Cauchy residue theorem as follows: Take a rectangle with vertices at s = c + it, - T < t < T, s = [sigma] + iT, - a < [sigma] < c, s = - a + it, - T < t < T and s = [sigma] - iT, - a < [sigma] < c, where T > 0 is to mean [T.sub.1] > 0 and [T.sub.2] > 0 tending to [infinity] independently but we usually use this convention. In order to find the residue by partial fractions, we would have to differentiate 16 times and then substitute 0 into our result. Orlando, FL: Academic Press, pp. Second, we will need to show that the second integral on the right goes to zero. integral is therefore given by. Cauchy’s Residue Theorem Dan Sloughter Furman University Mathematics 39 May 24, 2004 45.1 Cauchy’s residue theorem The following result, Cauchy’s residue theorem, follows from our previous work on integrals. 1.The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. So we will not need to generalize contour integrals to “improper contour integrals”. Cauchy’s residue theorem Cauchy’s residue theorem is a consequence of Cauchy’s integral formula f(z 0) = 1 2ˇi I C f(z) z z 0 dz; where fis an analytic function and Cis a simple closed contour in the complex plane enclosing the point z 0 with positive orientation which means that it is traversed counterclockwise. New York: Residues can and are very often used to evaluate real integrals encountered in physics and engineering whose evaluations are resisted by elementary techniques. Here are classical examples, before I show applications to kernel methods. theorem gives the general result. 1 2πi Z γ f(z) dz = Xn i=1 Res(f,zi) . All tip submissions are carefully reviewed before being published. Orlando, FL: Academic Press, pp. By signing up you are agreeing to receive emails according to our privacy policy. The Cauchy Residue theorem has wide application in many areas of pure and applied mathematics, it is a basic tool both in engineering mathematics and also in the purest parts of geometric analysis. 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. Using the contour This article has been viewed 14,716 times. Practice online or make a printable study sheet. Suppose that f(z) has an isolated singularity at z0 and f(z) = X∞ k=−∞ ak(z − z0)k is its Laurent expansion in a deleted neighbourhood of z0. Dover, pp. Let Ube a simply connected domain, and fz 1; ;z kg U. We will resolve Eq. The diagram above shows an example of the residue theorem … 2 CHAPTER 3. The 5 mistakes you'll probably make in your first relationship. Join the initiative for modernizing math education. the contour, which have residues of 0 and 2, respectively. Take ǫ so small that Di = {|z−zi| ≤ ǫ} are all disjoint and contained in D. Applying Cauchy’s theorem to the domain D \ Sn 1=1 Di leads to the above formula. The residue theorem implies I= 2ˇi X residues of finside the unit circle. f(x) = cos(x), g(z) = eiz. Cauchy’s residue theorem let Cbe a positively oriented simple closed contour Theorem: if fis analytic inside and on Cexcept for a nite number of singular points z 1;z 2;:::;z ninside C, then Z C f(z)dz= j2ˇ Xn k=1 Res z=zk f(z) Proof. residue. Proof. Z b a f(x)dx The general approach is always the same 1.Find a complex analytic function g(z) which either equals fon the real axis or which is closely connected to f, e.g. Er stellt eine Verallgemeinerung des cauchyschen Integralsatzes und der cauchyschen Integralformel dar. If C is a closed contour oriented counterclockwise lying entirely in D having the property that the region surrounded by C is a simply connected subdomain of D (i.e., if C is continuously deformable to a point) and a is inside C, then f(a)= 1 2πi C f(z) z −a dz. Active 1 year, 2 months ago. It is easy to apply the Cauchy integral formula to both terms. and then substitute these expressions for sin θ and cos θ as expressed in terms of z and z-1 into R 1 (sin θ, cos θ). https://mathworld.wolfram.com/ResidueTheorem.html. Preliminaries. Zeros to Tally Squarefree Divisors. (Cauchy’s integral formula)Suppose Cis a simple closed curve and the function f(z) is analytic on a region containing Cand its interior. Theorem 4.1. Cauchy’s theorem tells us that the integral of f (z) around any simple closed curve that doesn’t enclose any singular points is zero. : "Schaum's Outline of Complex Variables" by Murray Spiegel, Seymour Lipschutz, John Schiller, Dennis Spellman (Chapter $4$ ) (McGraw-Hill Education) A generalization of Cauchy’s theorem is the following residue theorem: Corollary 1.5 (The residue theorem) f ∈ Cω(D \{zi}n i=1), D open containing {zi} with boundary δD = γ. THE GENERAL CAUCHY THEOREM (b) Let R αbe the ray [0,eiα,∞)={reiα: r≥ 0}.The functions log and arg are continuous at each point of the “slit” complex planeC \ R α, and discontinuous at each pointofR α. We assume Cis oriented counterclockwise. By the general form of Cauchy’s theorem, Z f(z)dz= 0 , Z 1 f(z)dz= Z 2 f(z)dz+ I where I is the contribution from the two black horizontal segments separated by a distance . Theorem 22.1 (Cauchy Integral Formula). The discussion of the residue theorem is therefore limited here to that simplest form. THE GENERAL CAUCHY THEOREM (b) Let R αbe the ray [0,eiα,∞)={reiα: r≥ 0}.The functions log and arg are continuous at each point of the “slit” complex planeC \ R α, and discontinuous at each pointofR α. Residue theorem. Take ǫ so small that Di = {|z−zi| ≤ ǫ} are all disjoint and contained in D. Applying Cauchy’s theorem to the domain D \ Sn If f is analytic on and inside C except for the ﬁnite number of singular points z It is not currently accepting answers. 3.We will avoid situations where the function “blows up” (goes to inﬁnity) on the contour. 129-134, 1996. In this very short vignette, I will use contour integration to evaluate Z ∞ x=−∞ eix 1+x2 dx (1) using numerical methods. Get the free "Residue Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. 2. We know ads can be annoying, but they’re what allow us to make all of wikiHow available for free. (c)Thefunctionlog αisanalyticonC\R,anditsderivativeisgivenbylog α(z)=1/z. Proposition 1.1. 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. The Residue Theorem has the Cauchy-Goursat Theorem as a special case. See more examples in http://residuetheorem.com/, and many in [11]. If z is any point inside C, then f(n)(z)= n! 1 Residue theorem problems We will solve several problems using the following theorem: Theorem. From MathWorld--A Wolfram Web Resource. If f(z) is analytic inside and on C except at a ﬁnite number of isolated singularities z 1,z 2,...,z n, then C f(z)dz =2πi n j=1 Res(f;z j). §6.3 in Mathematical Methods for Physicists, 3rd ed. It generalizes the Cauchy integral theorem and Cauchy's integral formula.From a geometrical perspective, it is a special case of the generalized Stokes' theorem. Cauchy's Residue Theorem contradiction? Theorem 23.4 (Cauchy Integral Formula, General Version). First, the residue of the function, Then, we simply rewrite the denominator in terms of power series, multiply them out, and check the coefficient of the, The function has two poles at these locations. Include your email address to get a message when this question is answered. 1. The diagram above shows an example of the residue theorem applied to the illustrated contour and the function, Only the poles at 1 and are contained in X is holomorphic, and z0 2 U, then the function g(z)=f (z)/(z z0) is holomorphic on U \{z0},soforanysimple closed curve in U enclosing z0 the Residue Theorem gives 1 2⇡i ‰ f (z) z z0 dz = 1 2⇡i ‰ g(z) dz = Res(g, z0)I (,z0); 11.2.2 Axial Solution in the Physical Domain by Residue Theorem. Knowledge-based programming for everyone. Weisstein, Eric W. "Residue Theorem." In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. In this very short vignette, I will use contour integration to evaluate Z ∞ x=−∞ eix 1+x2 dx (1) using numerical methods. With the constraint. The residue theorem, sometimes called Cauchy's Residue Theorem [1], in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. Let Ube a simply connected domain, and let f: U!C be holomorphic. The proof is based on simple 'local' properties of analytic functions that can be derived from Cauchy's theorem for analytic functions on a disc, and it may be compared with the treatment in Ahlfors [l, pp. Corollary (Cauchy’s theorem for simply connected domains). The #1 tool for creating Demonstrations and anything technical. Well, it means you have rigorously proved a version that will cope with the main applications of the theorem: Cauchy’s residue theorem to evaluation of improper real integrals.