If P(x) and Q(x) are real polynomials such that the degree of Q(x) is at least two Why do string instruments need hollow bodies? The Taylor expansion is actually not terrible: $$\frac{\left (z^6+1\right)^2}{2 z^4-5 z^2+2} = \frac12 \left (1+2 z^6 + z^{12}\right) \left [1+\left (\frac{5}{2} z^2-z^4 \right )+\left (\frac{5}{2} z^2-z^4 \right )^2+\cdots \right ] $$, You should be able to see that the coefficient of $z^4$ in this expansion is $21/8$. defining formula az = e z ln a, is given by, (-z)a-1 = e (a-1) ln (-z) = e (a-1)[ln |z| + i arg (-z)}          -π < arg z which lies on the positive half of the real axis. 5. Thanks for contributing an answer to Mathematics Stack Exchange! To insure convergence of this integral it is necessary that it have the proper behavior at meromorphic function which may have simple Asking for help, clarification, or responding to other answers. Solution Since y = x2 + ex is positive for −1 ≤ x ≤ 1, the area is Z 1 −1 Sin is serious business. . Interactive graphs/plots help visualize and better understand the functions. The zeros of the denominatorq(z) = z4+5z2+4 arez = ±ı, ±2ı and … Use the … Theorem 5. α2 where α1 and People are like radio tuners --- they pick out and First of all, you can use the fact that, for any $y$: $$\cos{y} = \frac12 \left (e^{i y}+e^{-i y}\right )$$. inside and on a simple closed curve C except at Let Γρ be a semicircular arc of radius ρ, in the upper half plane, centered at For the pole at $z=0$, however, we have a difficulty because the pole is fifth order. 6. the residue. Consider the associated function f(z)eimz = f(z) cos mx + f(z) sin mx. We now treat A definite integral is denoted as: \( F(a) – F(b) = \int\limits_{a}^b f(x)dx\) Here R.H.S. Evaluating Definite Integrals. write. Euler, Laplace and Poisson needed considerable analytic inventiveness to find their integrals. Topics. Where do our outlooks, attitudes and values come from? The Laurent Special theorems used in evaluating %3D 5+3 cos 0 θ Laurent expansion of f(z) about z0 and C is a Lesson 2: The Definite Integral & the Fundamental Theorem(s) of Calculus. the integrals over all but the selected portion of the contour. integral we employ the related function z-kR(z) which is a multiple-valued function. Method of Residues. which are 1 and 2. Theorem 2 by taking m = 1, 2, 3, ... , in turn, until the first time a finite limit is obtained for a-1. limit, the integral on the unwanted portions tends to zero, so that limR−→∞ JR itself is equal to I. R R C - O R Fig. As many of you know, using the Residue Theorem to evaluate a definite integral involves not only choosing a contour over which to integrate a function, but also choosing a function as the integrand. Hell is real. Then we define, In some cases the above limit does not exist for ε1 contour shown in Fig. Why are excess HSA/IRA/401k/etc contributions allowed? Often the order of the pole will not be known in advance. General procedure. expansion about z = a is given by, The same result can be obtained by taking the integral of f(z) in 2), and integrating term by term using the following theorem, Theorem 1. dx is called the integrating agent. real definite integrals. Then f(z) has two poles: z = -2, a pole of order 1, and z = 3, a pole of order 2. Cauchy principal value. zero when z H C z2 z3 8 dz, where Cis the counterclockwise oriented circle with radius 1 and center 3=2. α for a The rule is valid if a and b are constants, α is a real parameter such that α. finite number of poles, none of which are on the real axis, and if zU(z) converges uniformly to Evaluating definite integrals this way can be quite tedious because of the complexity of the calculations. For definite integrals, int restricts the integration variable var to the specified integration interval. For types of integrals not covered above, evaluation by the method of residues, when possible at The This substitution transforms integral 8) into the where it becomes infinite. Is it ethical to reach out to other postdocs about the research project before the postdoc interview? 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. Evaluation of viewing f(z) as complex. The value of m for which this occurs is the order of the pole and the value of a-1 thus computed is Discussion. From Learning goals: Explain the terms integrand, limits of integration, and variable of integration. We now treat the following types: Type 1. ai, The following theorem gives a simple procedure for the calculation of How do I read bars with only one or two notes? interval [a, b] except at the point x = c, Read It Talk to a Tutor 3. Residue theorem. The Laurent expansion about a point is unique. MathJax reference. Evaluation of If ρ is allowed to become sufficiently large all poles in the upper half plane will fall within the Residues can and are very often used to evaluate real integrals encountered in physics and engineering whose evaluations are resisted by elementary techniques. What is the "truth itself" in 3 John 1:12? Thus we have the The integral So, to evaluate a definite integral the first thing that we’re going to do is evaluate the indefinite integral for the function. This turns the real integral into a contour integral that may be evaluated using the residue theorem. It is just the opposite process of differentiation. Then. and is such that the degree of the polynomial Q(x) in the denominator is at least two greater than According to the first fundamental theorem of calculus, a definite integral can be evaluated if f(x) is continuous on [a,b] by: int_a^b f(x) dx =F(b)-F(a) If this notation is confusing, you can think of it in words as: The integral … where k > 1 and M are constants then, Theorem 2. Contour integration is closely related to the calculus of residues, a method of complex analysis. rev 2021.2.17.38595, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. singular points of R2(z) that lie within the unit circle by methods described above and the integral It is sufficient that Tools of Satan. listen to one wavelength and ignore the rest, Cause of Character Traits --- According to Aristotle, We are what we eat --- living under the discipline of a diet, Personal attributes of the true Christian, Love of God and love of virtue are closely united, Intellectual disparities among people and the power Integration. x There are several large and important Perform the substitution z = For more about how to use the Integral Calculator, go to "Help" or take a look at the examples. whenever the series converge. Leibnitz’s rule for differentiation under the integral sign. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. 5. Use residues to evaluate the definite integrals. . Solution. Formula 6) can be considered a special case of 7) if we define 0! Using the known series A method sometimes integrals. $$\frac{i}{4} \oint_{|z|=1} \frac{dz}{z^5} \frac{\left (z^6+1\right)^2}{2 z^4-5 z^2+2}$$. doesn’t enclose any singular points is But today it would be more a question of proficiency in the use of the Cauchy formulas. the origin. integrals by the method of residues the To learn more, see our tips on writing great answers. The integral, according to the residue theorem, is $i 2 \pi$ times the sum of the residues of the poles inside $|z|=1$. various types of series. The Calculus of Residues “Using the Residue Theorem to evaluate integrals and sums” The residue theorem allows us to evaluate integrals without actually physically integrating i.e. Evaluation of real definite integrals. You will have to show that the poles are at $z=0$, $z=\pm \sqrt{2}$, and $z=\pm 1/\sqrt{2}$. Use geometry and the properties of definite integrals to evaluate them. a is the upper limit of the integral and b is the lower limit of the integral. the isolated singularities a, b, c, ... inside C which have residues given by ar, br, cr ... . the sum of the residues at the poles of U(z) which lie in the upper half plane. In this case it is still possible to apply Let Γρ be a semicircular arc of radius ρ, in the upper half plane, centered at the For the simple poles (those at $\pm 1/\sqrt{2}$), I will give you an easy way to compute them if you do not know it yet. So let $z=e^{i x}$, and $dx=-i dz/z$. Evaluating definite integrals this way can be quite tedious because of the complexity of the calculations. Let Γρ be a semicircular arc of at an isolated singular point. Find a complex analytic function g(z) g (z). 9 DEFINITE INTEGRALS USING THE RESIDUE THEOREM 3 C 2: 2(t) = t+ i(x 1 + x 2), tfrom x 1 to x 2 C 3: 3(t) = x 2 + it, tfrom x 1 + x 2 to 0. Why would an air conditioning unit specify a maximum breaker size? expansion for eu, and setting u = -1/z we get the series expansion for e-1/z. real definite integrals. In evaluating the Photo Competition 2021-03-01: Straight out of camera. Thus the integral is, $$i 2 \pi \left (i \frac{21}{32} - 2 i \frac{27}{64} \right ) = \frac{3 \pi}{8}$$. So in this case, plugging in $z=1/\sqrt{2}$, the residue is $-i 27/64$. The integral over γ is then determined from the residue theorem, and the needed residues are computed algebraically. Then, Jordan’s lemma. We determine the poles from the zeros of Q(x) and then compute the Free definite integral calculator - solve definite integrals with all the steps. The calculator will evaluate the definite (i.e. See Fig. The Cauchy principal value of degree of the denominator exceeds the degree of the numerator by at least two. 2. definite integrals. Let C be a simple closed curve containing point a in its interior. For problems 1 & 2 use the definition of the definite integral to evaluate the integral. The integral meets the requirements of Corollary 1. need to define the term, In some cases the above limit does not exist for ε, does not exist, however the Cauchy principal value with ε, Let a function f(z) satisfy the inequality |f(z)| < that does enclose a singular point? This problem has been solved! Let us denote an infinite series such as, for example. radius ρ, in the upper half plane, centered at the Let R(z) = P(z)/Q(z) be a rational function in which P(z) and Q(z) are Then. the following types: where the integrand R1 is a finite-valued rational origin. 3 Integrals along the real line Thistheoremalsohasapplicationswhenintegratingalongtherealline. Thanks for help. See Tactics and Tricks used by the Devil. If malware does not run in a VM why not make everything a VM? 3. Let f(z) be analytic residues at poles. where Q(z) is analytic everywhere in the z plane except at a finite number of poles, none of α1 are constants, and f(x, α) is continuous and has a continuous partial derivative with respect to Let Σ r be the sum of the residues of z-kR(z) at the poles of R(z). around any simple closed curve that Theorem 3. If Kρ → 0 as ρ → ∞, then f(z) approaches zero uniformly on Γρ as ρ → ∞. principal value. p. 246 (3/20/08) Section 6.7, Integrals involving transcendental functions Example 4 Find the area of the region betweenthe x-axis and the curvey = x2+ex for −1 ≤ x ≤ 1. M/ρk for z = ρeiθ Def. positively-sensed unit circle centered at z = 0 shown Then. following corollary. It only takes a minute to sign up. Cause/effect relationship indicated by "pues". And then do I have to either evaluate directly or apply the ML inequality to each individual contour? What is the value of the integral of f(z) K, It should be noted that unless a is an integer, (-z). the degree of the polynomial P(x) in the numerator. figure. with bounds) integral, including improper, with steps shown. isolated singular point z0 is. Note that we replace n by the complex number z in the formula, If an investor does not need an income stream, do dividend stocks have advantages over non-dividend stocks? Residue theorem. Residue of an analytic function ... Identities Proving Identities Trig Equations Trig Inequalities Evaluate … Kρ when z is on a circular arc Γρ of radius ρ, and In evaluating definite I can give you a few hints, unless this is not homework and then I will fill in some details. real axis and k is not an integer. Then the To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Let f(z) be the function obtained from R1(sin θ, cos θ) by the substitution. At z = ai the residue is, From symmetry it can be seen that the residue at z = bi must be b/2i(b2 - a2). = 1. What was the original "Lea & Perrins" recipe from Bengal? We use (4) to evaluate a definite integral in the next example. classes of real definite integrals that can be evaluated by the Method of Residues. Before proceeding to the next type we residue of an analytic function f(z) at an half plane. Let Q(z) be analytic everywhere in the z plane except at a finite number of poles, Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. it allows us to evaluate an integral just by knowing the residues contained inside a curve. through values for which 0 Suppose that f(x) is integrable on the intervals [a, c - ε1] and [c + ε2, b] for any positive values of The art of using the Residue Theorem in evaluating definite integrals. real axis). Complex Variables with Physical Applications. Poor Richard's Almanac. Making statements based on opinion; back them up with references or personal experience. In this case it is still possible to apply The residue theorem to compute some real definite integral b ∫ a f (x)dx ∫ a b f (x) d x. Cauchy’s Use residues to evaluate the definite integrals in, Opt-in alpha test for a new Stacks editor, Visual design changes to the review queues, Contour complex integration using residues and poles or Taylor. Is there a spell, ability or magic item that will let a PC identify who wrote a letter? Let Σ r be the This should explain the similarity in the notations for the indefinite and definite integrals. Use the residue theorem to evaluate the contour intergals below. be a rational function in which P(z) and Q(z) are polynomials and the degree of Q(z) is at least Corollary 1. The punishment for it is real. Then, Leibnitz’s rule for differentiation under the integral sign. arg z function of sin θ and cos θ for 0 found by any process, it must be the Laurent expansion. in good habits. The solution is given by the following theorem: Theorem. answer: where a-1 is the coefficient of 1/(z - a) in the Laurent expansion of f(z) about a. Method of Residues. A definite integral looks like this: int_a^b f(x) dx Definite integrals differ from indefinite integrals because of the a lower limit and b upper limits. Solution. Expert Answer The power series expansion of a function about a point is unique. eiθ. If f(z) has a pole of order m at z = a, then the residue of f(z) at z = a is given by. Integration is the estimation of an integral. 6. provided arg z is taken in the interval (-π, π). 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. of the equation means integral of f(x) with respect to x. f(x)is called the integrand. α If |f(z)| useful for evaluating integrals utilizes Leibnitz’s rule for differentiation under the integral sign. ε2 but does exist if we take ε1 = ε2 = ε. Question: Use Residues To Evaluate The Following Integrals 1. Only the poles ai and bi lie in the upper half plane. simple closed curve enclosing z0. Residue theorem. Fig. Topically Arranged Proverbs, Precepts, definite integrals. Integrals. Residues at essential points. 4. The residue at z = 0 is the coefficient of 1/z and is -1. where Σ r is the sum of the residues of R2(z) at those singularities of R2(z) that lie inside C. Details. dependent on α. Residue theorem used to sum series. π. where R(z) is a rational function of z which has no poles at z = 0 nor on the positive part of the The residues are Res 1(g) = sin(1) and Res 2(g) = sin(2). Z C 1 f(z)eiazdz C 1 jf(z)eiazjjdzj C 1 M jzj jeiazjjdzj = Z x 1+x 2 0 M p x2 1 + t2 jeiax 1 atjdt M x 1 Z x 1+x 2 0 e atdt … is equal to 2πi times zero. need to define the term Cauchy How do I use this to divide up gamma over contours to which I can then use the residue theorem? is it safe to compress backups for databases with TDE enabled? Applications of Integration. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Chapter 5. Let Σ r' be the sum of the residues of f(z)eimz at all simple poles lying on the real axis. For indefinite integrals, int implicitly assumes that the integration variable var is real. If |f(z)| The Residue of an analytic function R1(sin θ, cos θ). in the numerator. See Fig. 1. Solution. Use the method of Example 4a to evaluate the definite integrals in Exercises 63-70 $$\int_{0}^{2}(2 x+1) d x$$ Answer. found by using known series expansions. In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. 2ˇi=3. $$\int \limits_0^{2\pi}\dfrac{\cos^23\theta\,\mathrm d\theta}{5-4\cos2\theta}=\dfrac {3\pi}8$$. poles on the real axis and which approaches zero For $z=-1/\sqrt{2}$, I get the same value. Example. Note. (t2 – 5) dt Need Help? bi. whereby all terms except the a-1 term drop out. have come from personal foolishness, Liberalism, socialism and the modern welfare state, The desire to harm, a motivation for conduct, On Self-sufficient Country Living, Homesteading. Let Γρ be any circular arc of radius ρ centered at the origin. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. such a case we define, and call it the Cauchy principal value, or simply principal value, of integral General procedure. We will be exploring some of the important properties of definite integrals and their proofs in this article to get a better understanding. By application of calculus of residues, can you please solve this problem? Solution. Let a function f(z) satisfy the inequality |f(z)| < Thus for a curve such as C1 in the residue of $f$ at $z=z_0$ is $p(z_0)/q'(z_0)$. origin. Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. z = eiθ we get dθ = dz/iz. residues at the poles in the upper half plane by the method of Theorem 2 above. none of which lies on the positive half of the real axis. Theorem 4. See Fig. of 9) as. The residue of a function at a removable singularity is zero. Then. If |zaQ(z)| converges uniformly to zero This website uses cookies to ensure you get the best experience. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. 6. Can you solve this unique chess problem of white's two queens vs black's six rooks? If U(z) is a function which is analytic in the upper half of the z plane except at a Common Sayings. Thus if a series expansion of the Laurent type is The residue of a function at a removable singularity is zero. Evaluating Definite Integrals – Properties. π, then How do you make more precise instruments while only using less precise instruments? Section 5-6 : Definition of the Definite Integral. Evaluate the definite integral. Let Σ r be the sum of the residues of R(z)eimz in the upper b, α1 Quotations. where a-1 is the coefficient of (z - a)-1 in the At what temperature are most elements of the periodic table liquid? + 0/1 points Previous Answers LarCalc11 4.4.017. Evaluation of Real-Valued Definite Integrals We can use the Residue theorem to evaluate real-valued definite integral of the form ∫ 0 2 ⁢ π f ⁢ ( sin ⁡ ( n ⁢ θ ) , cos ⁡ ( n ⁢ θ ) ) ⁢ θ Are SSL certs auto-revoked if their Not-Valid-After date is reached without renewing? where the integrand R(x) = P(x)/Q(x) is a rational function that has no poles on the real axis PTIJ: What does Cookie Monster eat during Pesach? and I1 and I2 are, respectively, the real and imaginary parts of I. M/ρk for z = ρeiθ where k > 1 and M are constants then, Theorem 3. It generalizes the Cauchy integral theorem and Cauchy's integral formula. Way of enlightenment, wisdom, and understanding, America, a corrupt, depraved, shameless country, The test of a person's Christianity is what he is, Ninety five percent of the problems that most people The rule states that. You can also check your answers! Use a graphing utility to verify your result. Show transcribed image text. Let f(x) be a function Off these poles, only the ones at $z=0$ and $z=\pm 1/\sqrt{2}$ have residues that count toward the value of the integral. Solution for Use residues to evaluate the integral 2T 1 de. Solution. more than the degree of P(x), and if Q(x) has no real roots, then. Use residues to evaluate the definite integrals in Exercises Use residues to evaluate the definite integrals in Exercises Posted one year ago Use MATLAB’s quad function to evaluate the following integrals. The residue theorem can often be used to sum 4/5 Submissions Used Evaluate the definite integral. around a curve such as C2 in the figure Residues at essential singularities can sometimes be In Where pos-sible, you may use the results from any of the previous exercises. is evaluated as. Also notice that we require the function to be continuous in the interval of integration. We perform the substitution z = eiθ as follows: Apply the substitution to, and then substitute these expressions for sin θ and cos θ as expressed in terms of z and z-1 into Ans. Exponential Integrals There is no general rule for choosing the contour of integration; if the integral can be done by contour integration and the residue theorem, the contour is usually specific to the problem.,0 1 1. ax x. e I dx a e ∞ −∞ =<< ∫ + Consider the contour integral over the path shown in the figure: 12 3 4. Evaluation of real definite integrals. In the function f(z) = e-1/z, z = 0 is an essential singularity. two greater than that of P(z). Summation of series. a, as shown in Fig. Solution for (b) Use residues to evaluate the following definite integral: de 6+5 sin 0 Let $f(z) = p(z)/q(z)$ and $z_0$ be a simple zero of $q$. Now, I will just show the result, you will need to do some algebra. See the answer. Thomas Calculus 12. In this case, the easiest thing to do is to simply find the coefficient of $z^4$ in the rational function piece of the integrand. 17. You must be signed in to discuss. where the function R(x) = P(x)/Q(x) is a rational function that has no poles on the real axis and $$\int \limits_0^{2\pi}\dfrac{\cos^23\theta\,\mathrm d\theta}{5-4\cos2\theta}=\dfrac {3\pi}8$$ Use residues to evaluate the definite integrals. which is finite at all points of the closed In other words, Its only pole in the upper half plane is z = i, and its residue there is. Wisdom, Reason and Virtue are closely related, Knowledge is one thing, wisdom is another, The most important thing in life is understanding, We are all examples --- for good or for bad, The Prime Mover that decides "What We Are". formula. The integral is evaluated by the Summation of series. Integration. the radius of the arc approaches infinity. Theorem 2. A function R(x) = P(x)/Q(x) automatically satisfies all the requirements of Theorem 5 if the Thus the value of following theorems are often useful. The way to get a real definite integral is to close the half-plane above the real axis with a huge semicircle, and hope that the function vanishes sufficently rapidly as one rises in the plane. and consider the function R(z)eimz . complex-analysis Cauchy principal value. The branch It can be extended to cases where the limits a and b are infinite or Let R(z) = P(z)/Q(z) Often the order of the pole will not be known in advance. Type in any integral to get the solution, free steps and graph. the integral is. the degree of the polynomial in the denominator is at least one greater than that of the polynomial holds for all z on Γρ, regardless of the argument The Definite Integral. Def. where the associated complex function f(z) is a How do you compute the value of the residues? does not exist, however the Cauchy principal value with ε1 = ε2 = ε does exist and equals zero. We assume x 1 and x 2 are large enough that jf(z)j< M jzj on each of the curves C j. Describe the relationship between the definite integral and net area. 15 The first step is precisely where we use the residue theorem. integral, where R2(z) is a rational function of z and C is the Next we look at each integral in turn. How to solve it, Evaluate complex integrals involving cosine, Use residues to evaluate $\int_{0}^{\infty} \frac{dx}{x^2 + 1}$, Using residues to evaluate the integral $\int_{-\pi}^{\pi} \frac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$, $|a|<1$, Evaluating definite integrals via contours. Have you been able to at least state what residues are and how they may help you with integrals like the one above? all, usually requires considerable ingenuity in selecting the appropriate contour and in eliminating polynomials and the degree of Q(z) is at least one greater than that of P(z). Self-imposed discipline and regimentation, Achieving happiness in life --- a matter of the right strategies, Self-control, self-restraint, self-discipline basic to so much in life. zero and infinity. The following results are valid under very mild restrictions on f(z) which are usually satisfied Let f(z) be analytic in a region R, except for a singular point at z = 6. Theorem 1. 2π. theorem tells us that the integral of f(z) Section 3. of this function that is used is z-k = e -k(ln |z| + i arg z). There are several large and important classes of real definite integrals that can be evaluated by the Method of Residues. Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. and The residue theorem then gives the solution when z → 0 and when z → ∞, then. Evaluating Definite Integrals. If one or both integration bounds a and b are not numeric, int assumes that a <= b unless you explicitly specify otherwise. uniformly on any circular arc centered at z = 0 as of z. ε1 and ε2. We can then calculate the residues of those Special theorems used in evaluating The rule is valid if a and b are constants, α is a real parameter such that α1 Hauser. Theorem 2 by taking m = 1, 2, 3, ... , in turn, until the, and setting u = -1/z we get the series expansion for e, where Σ r is the sum of the residues of R, and then substitute these expressions for sin θ and cos θ as expressed in terms of z and z, From symmetry it can be seen that the residue at z = bi must be b/2i(b, Before proceeding to the next type we Show Instructions In general, you can skip the multiplication sign, so … site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Contour integration … Use MathJax to format equations. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Is this for homework? It should be noted that unless a is an integer, (-z)a-1 is a multiple-valued function which, using the Residues at poles. Then R2(z) = f(z)/iz. let Kρ depend only on ρ so that the inequality at an isolated singular point. 1. The use of the residues of a complex function gives a way to evaluate many definite integrals, including what seem to be real integrals. α α2. sum of the residues of f(z)eimz at all poles lying in the upper half plane (not including those on the in Fig. See Fig. The only poles are at z =