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## Complex Analysis : Complex Integration

Study concepts, example questions & explanations for complex analysis, all complex analysis resources, example questions, example question #41 : complex analysis.

Recall Cauchy's integral formula, which states

Plugging in gives us

## Example Question #1 : Complex Integration

Recall the Taylor expansion of

Use this to write

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## Complex Integration

Complex integration is a simple extension of the ideas we develop in calculus to the complex world. In real calculus, differentiation and integration are, roughly speaking, inverse operations (save for the additional interpretation of derivative as the slope of a function and integral as the area under the curve). We will see that the same inverse relationship between differentiation and integration exists in the complex domain; but in addition, differentiation and integration are also roughly equivalent operations. But to see this striking result, we have to wait till Cauchy’s Theorem. This article grows over the basics and is necessary for preparation for fully appreciating Cauchy’s Theorem and its results.

## Complex integration of a real variable

Start with the basic, that is, for a complex function of a real variable, and then slowly ramp up to more complicated, yet exciting, cases of complex integration. Suppose that f is a complex-valued function of a real variable t. We can write f(t) = u(t) + iv(t), where u and v are now real-valued functions of a real variable. We can therefore define the integral of f(t) on the interval [a, b] as

Example 1: Integrate f(t) = 2t + it 2 . Here, u(t) = 2t and v(t) = t 2 . So, for 0≤t≤1.

Example 2: Solve f(t) = ie t . Integrating it from 0 to 2.

## Contour Integrals

Where z 0 and z 1 are complex numbers. We immediately see a difficulty. Unlike the case of an interval [a, b] where it is obvious how to go from a to b, here we have points in the complex plane and there are infinitely many ways to go from one point to the other since the complex plane is a 2-dimensional plane. Therefore, the above integral is ambiguous. To do this integration, we have to define a path from z o to z 1 . This is similar to line integrals of vector functions in 3-dimensional space, where we need a specific parameterized path to compute the integral.

Definition: Let z 0 and z 1 be two points in the complex plane. A parameterized curve joining z 0 and z 1 can be defined by a continuous function z: [t 0 , t 1 ] → C such that z(t 0 ) = z 0 and z(t 1 ) = z 1 . We can think of t here as time. At any given instant in time t 0 ≤ t ≤ t 1 , the particle is at the point z(t) in the complex plane. Therefore we see that a curve joining z 0 and z 1 can be defined by a function z(t) mapping points t-domain, in the interval [t 0 , t 1 ], to corresponding points z(t) in the complex plane, such that z(t 0 ) = z 0 and z(t 1 ) = z 1 . As an immediate result of this definition, we can decompose z into its real and imaginary parts, and this is equivalent to two continuous real-valued functions x(t) and y(t) defined on the interval [t 0 , t 1 ] such that x(t 0 ) = x 0 and x(t 1 ) = x 1 and similarly for y(t): y(t 0 ) = y 0 and y(t 1 ) = y 1 , where z 0 = x 0 + iy 0 and z 1 = x 1 + iy 1 . We call the curve smooth if its velocity dz(t)/dt = z'(t) is a continuous function in the interval [t 0 , t 1 ] which is never zero.

Let Γ be a smooth parameterized curve joining z 0 to z 1 , and let f(z) be a complex-valued function that is continuous on Γ. Then we define the integral of f along Γ by,

The complex integral also satisfies some properties like linearity and change of sign when integrated into the reverse direction. The proofs of these properties are easy and can be easily done by using the definition.

Linearity property

∫ Γ (αf(z) + βg(z))dz = α∫ Γ f(z)dz + β∫ Γ g(z))dz

Sign reversal property

∫ −Γ f(z)dz = −∫ Γ f(z)dz

Example 1: Consider the function f(z) = |z| 2 = x 2 + y 2 integrated along the curve parameterized by z(t) = t + it for 0 ≤ t ≤ 1.

This is the straight line segment joining the origin and the point 1+ i. We have here ̇dz(t)/dt = z'(t) = 1 + i. So,

Example 2: Consider now the function f(z) = 1/z integrated along the smooth curve Γ parameterized by z(θ) = R e (i θ) for 0 ≤ θ ≤ 2π, where R ≠ 0.

The curve is a circle of radius R centered about the origin. Here f(z(θ)) = (1/R) e (−i θ) and dz(θ)/dθ = z'(θ) = i R e (i θ) = iz. So, we have: Note that the result is independent of R.

Example 3: Compute the integral of ln(z) over the unit circle |z| = 1. Parameterized z = e iθ .

So, considering 0 ≤ θ ≤ 2π. Here we took |z| = 1 as it is specified. Now, dz(θ)/dθ = z'(θ) = ie iθ . After doing integration by parts, we finally get 2πi.

Results from Contour Integration

Result 1: Take f(z) = 1, then according to the above definition,

Result 2: The length of the curve is the contour integral of the modulus of dz, the real function |dz|. The length of the curve Γ is, therefore,

Result 3: Estimating the maximum value of a contour integral.

Piecewise-smooth curve

If Γ 1 is a smooth curve joining z 0 to z 1 and Γ 2 is another smooth curve joining z 1 to z 2 , then we can make a curve Γ, not necessarily smooth, by joining z 0 to z 2 by first going to the intermediate point z 1 via Γ 1 and then from there via Γ 2 to our destination z 2 . The resulting curve Γ is still continuous, but it may not be smooth, since the velocity need not be continuous at the intermediate point z 1 . Such curves can be called piecewise smooth or contours, as popularly called in the literature. We can, thus, construct curves that are not smooth but which can be made out of a finite number of smooth curves, called smooth components.

Let Γ be a contour with n smooth components {Γ j } for j = 1, 2, . . . , n. If f(z) is a function continuous on Γ, then the contour integral of f along Γ is defined as

First task is to find |f(z)|. This is: |f(z)| = Line C is shown here. Equation of C is 2y + x – 2 = 0. Here, on the line C, |f(z)| becomes a function of one variable. Putting y = (2 – x)/2, we get, The maximum value attained by |f(z)| is e 4/5 at x = 4/5. Now the length of the contour C is basically the distance between (0,1) and (2,0), that is, √5. Using Result-3, we have I ≤ √5e 4/5

Example 2: Integrate f(z) = |z| 2 again, but over piecewise smooth contour.

Break the whole integration process into four parts. For OA, dy = 0. So dz = dx. y = 0, so |z| 2 = x 2 . For AB, dx = 0. So dz = idy. x = 1 so |z| 2 = 1+y 2 . For BC, dy = 0, So dz = dx. y = 1 so |z| 2 = x 2 +1. For CO, dx = 0, so dz = idy. x = 0, so |z| 2 = y 2 .

Path Independence Theorem

In vector calculus, the line integral becomes independent of the path in some special cases. This happens when the vector function can be expressed as the gradient of a scalar function. Similar is the notion of independence of path in complex integration. If we have a function whose anti-derivative exists in a region, we can say that the integral of the function will be independent of the path of integration. A domain is a subset (or a region) of a complex plane, in which every pair of points can be connected via a contour (or a piecewise-smooth curve).

Let D be a domain and let f: D → C be a continuous complex-valued function defined on D. We say that f has an antiderivative in D if there exists some function F: D → C such that,

Clearly, F is analytic in D. Let Γ be any contour in D with endpoints z 0 and z 1 . If f has an antiderivative F on D, the contour integral is given by, ∫ Γ f(z)dz = F(z 1 ) − F(z 0 ). The proof is straightforward. Start with the definition and parameterize Γ by z(t) for 0 ≤ t ≤ 1. Then,

For a more general contour, we can segment the whole contour into individual smooth components {Γj} for j = 1, 2, . . . , n. Then, we can readily see that the end points of each smooth component cancel out, leaving behind the start and end points.

∫ Γ f(z)dz = j = 1 ∑ n ∫ Γj f(z)dz = ∫ Γ1 f(z)dz + ∫ Γ2 f(z)dz + …. + ∫ Γn f(z)dz = F(τ 1 ) − F(z 0 ) + F(τ 2 ) − F(τ 1 )+….+F(z 1 ) − F(τ n−1 ) = F(z 1 ) − F(z 0 )

This result says that if a function f has an antiderivative, then its contour integrals do not depend on the precise path, but only on the endpoints. If we now have a closed contour, meaning the start and end points coincide, then the closed contour integral in such a case is zero. Explicitly stated, if Γ is a closed contour in some domain D and f : D → C has an antiderivative in D, then,

This is the precise mathematical statement of path independence of a complex function, whose anti-derivative exists in the given domain. For example, complex exponential and polynomials are functions whose anti-derivative exists and so their closed contour integrals are zero by the path-independence theorem.

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## How to Solve Complex Integration by Theorems

There are various kinds of methods of solving complex integrations, such as using the Cauchy’s integral theorem, the residue theorem or specifying a specific integration route. It is necessary to use the appropriate method for a complex integrations you want to solve. Of course, you should remember as many kinds of the methods as possible.

Reference: Cauchy-Riemann Equation and Holomorphic Function

## Method of Using Cauchy’s Integral Theorem

Cauchy’s integral theorem.

If the complex integral route \(C\) is a single closed curve and the function \(f(z)\) is holomorphic on the path \(C\) and its inside, the following expression holds.

$$\int_{C} f(z) dz=0$$

Calculate the complex integral:

$$\int_{|z|=1} z^3dz$$

## What is a complex plane?

This integration needs to be considered on a complex plane. A complex plane is the plane coordinate with the real part of the complex number on the horizontal axis and the imaginary part on the vertical axis. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

Here, \( |z|=1\) on the complex plane represents a circle with a radius of \(1\) centered at the origin \(|z|=0\). Defining \(z\) as \(z≡x+iy\), the following equation is obtained by the Pythagorean theorem and the complex plane.

$$|z|=\sqrt{x^{2}+y^{2}}･･･(1)$$

Substitute \(z=1\) into the equation (1).

$$x^{2}+y^{2}=1$$

This equation obviously expresses a circle on the plane coordinate.

Using the Cauchy-Riemann equation to \(f(z)=z^3\), the function \(f(z)\) is holomorphic. Therefore, the complex integral is caluculated by the Cauchy’s integral theorem.

$$\int_{|z|=1} z^{3} dz=0$$

## Method of Using Residue Theorem

Overview of laurent expansion.

The Laurent expansion is what extend the Taylor expansion to negative powers.

$$f(z)=\sum_{n=-∞}^{∞} {a_{n}(z-α)^{n}}$$

## Overview of Residue

The residue is the coefficient of \((z – α)^{- 1}\) of the Laurent series of \(f(x)\), that is, \(a_{-1}\) in the above equation of the Laurent expansion.

The residue of the function \(f(z)\) at \(z=α\) is expressed as

$$a_{-1}=Res(α,f(z)).$$

## Residue Theorem

The residue theorem is expressed as

$$\int_{C} f(z) dz=2πi\sum_{j=1}^{k}Res(α_{j},f(z)).$$

Here, \(k\) represents the number of non-holomorphic points that the function \(f(z)\) has on the integral route \(C\) or its inside.

## How to Obtain the Residue

In the previous section, we considered the case which the integral route and its interior are holomorphic with the Cauchy’s integral theorem. However, what if the integral route and its interior includes non-holomorphic points? In this case, the complex integrals can be obtained by the residue of \(f(z)\) at non-holomorphic points.

The residue \(Res(α,f)\) of the pole of order \(n\) of the function \(f(x)\) at \(z=α\) can be expressed as

$$Res(α,f)=\frac{1}{(n-1)!}\displaystyle \lim_{z \to α} \frac{d^{n-1}}{dz^{n-1}}(z-α)^{n}f(z).$$

For example, the residue \(Res(α,f)\) of the pole \(z=α\) of the order \(1\) of the function \ (f (z) \) expresses as

$$Res(α,f)=\displaystyle \lim_{z \to α} (z-α)f(z).$$

$$(i) \ \int_{|z|=1} \frac{1}{z}dz$$

$$(ii) \ \int_{|z-5i|=1} \frac{1}{z}dz$$

(i)\(|z|=1\)

The non-holomorphic point of \(f(z)=\frac{1}{z}\) is only \(z=0\). Then \(z=0\) is the point inside the integration route \(|z|=1\). In addition, the residue is obviously \(1\).

Substitution the residue into the residue theorem to calculate the integration.

$$\int_{|z|=1} \frac{1}{z} dz=2πiRes(0,\frac{1}{z})=2πi･1=2πi$$

(ii)\(|z-5i|=1\)

\(|z-5i|=1\) is a circle whose center is at \((x, y) = (0, 5)\) on the complex plane and its radius is \(1\). The non-holomorphic point of \(f(z)=\frac{1}{z}\) is only \(z=0\), but this point is out of the circle \(|z-5i|=1\). Therefore, the function \(f(z)\) is holomorphic on this integral route and its inside, and the complex integration is 0.

$$\int_{|z-5i|=1} \frac{1}{z} dz=0$$

$$\int_{|z+i|=1}f(z)=\frac{1}{z^{2}+1}$$

The function \(f(z)\) can be transformed as

$$f(z)=\frac{1}{z^{2}+1}=\frac{1}{(z+i)(z-i)}.$$

From this equation, there are two points \(f(z)\) is not holomorphic, that is, \(z=\pm i \). However, the point \(z=+i\) is out of the integration route \(|z+i|=1\), so it does not affect the integration result. Therefore, you only need to pay attention to the residues of the point \(z=-i\) within the integral route.

Using the formula for finding the residues above, find the residue for point \(z=-i\).

$$Res(-i,\frac{1}{z^{2}+1})=\displaystyle \lim_{z \to -i} (z+i)･\frac{1}{(z-i)(z+i)}=\displaystyle \lim_{z \to -i} \frac{1}{z-i}=-\frac{1}{2i}$$

From the above, the complex integration is obtained as follows.

$$\int_{|z+i|=1} \frac{1}{z^{2}+1} dz=2πi \left[ Res(-i,\frac{1}{z^{2}+1}) \right]=-2πi･\frac{1}{2i}=-π$$

## Method of Setting an Integration Route

If you can set the integration route yourself, you become able to solve more kinds of complex integrations. For example, the real integration that can not be integrated with elementary functions can be extended to the complex integration and solved.

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Many challenging integration problems can be solved surprisingly quickly by simply knowing the right technique to apply. While finding the right technique can be a matter of ingenuity, there are a dozen or so techniques that permit a more comprehensive approach to solving definite integrals.

Manipulations of definite integrals may rely upon specific limits for the integral, like with odd and even functions , or they may require directly changing the integrand itself, through some type of substitution . However, most integrals require a combination of techniques, and many of the more complicated approaches, like interpretation as a double integral , require multiple steps to reduce the expression.

Consider, for instance, the antiderivative

\[\displaystyle\int e^{- x^2} \, dx.\]

This is known as the Gaussian integral, after its usage in the Gaussian distribution , and it is well known to have no closed form. However, the improper integral

\[I = \int_0^\infty e^{- x^2} \, dx\]

may be evaluated precisely, using an integration trick . In fact, its value is given by the polar integral

\[I^2 = \int_0^\infty \int_0^\infty e^{-x^2} e^{-y^2} \, dy\, dx = \int_0^{\pi/2} \int_0^\infty r e^{-r^2} \, dr\, d\theta.\]

Without such a method for exact evaluation of the integral, the Gaussian (normal) distribution would be significantly more complicated. Such integrals appear throughout physics , statistics , and mathematics .

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## . Read the following definition of a Complex Engineering Problem...

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With the provided Information in question I have given perfect response Just go through the answer provided below Hope this helps Thank you !

1.The definition outlines several characteristics that contribute to the complexity of an engineering problem. If a design or problem exhibits one or more of the following traits, it can be considered complex:

Wide-ranging or conflicting technical issues: If the design involves diverse and conflicting technical requirements or considerations, it adds to the complexity. For example, designing a sustainable and energy-efficient building may involve conflicting requirements for insulation and ventilation.

No obvious solution: If there is no straightforward or readily apparent solution to the problem, it adds complexity. An example could be developing a cost-effective method for large-scale carbon capture and storage.

Addressing problems not encompassed by current standards and codes: If the design challenge goes beyond existing industry standards and codes, it introduces complexity. For instance, designing a novel medical device may require addressing regulatory challenges not covered by existing standards.

Involving diverse groups of stakeholders: If the design process requires collaboration and coordination among various stakeholders with different interests, it adds complexity. Designing a new urban transportation system involves stakeholders such as city planners, environmentalists, and transportation authorities.

Many component parts or sub-problems: If the design involves numerous interconnected components or sub-problems, it adds to the complexity. Designing a spacecraft involves addressing challenges related to propulsion, life support systems, navigation, and more.

Involving multiple disciplines: If the design requires expertise from multiple engineering disciplines, it adds complexity. An example could be designing a smart city, which involves expertise in civil engineering, electrical engineering, computer science, and more.

Having significant consequences in a range of contexts: If the design's outcome has widespread and profound consequences, it adds to the complexity. Designing a resilient and secure critical infrastructure system, such as a power grid, falls into this category.

if a given design exhibits one or more of these characteristics, it can be considered a complex engineering problem.

2.Complex Digital Solution Example: A problem that would require a complex digital solution is the development of a highly efficient and adaptive traffic management system for a smart city.

Wide-ranging or conflicting technical issues: Integrating real-time data from sensors, optimizing traffic flow, and ensuring data security present conflicting technical challenges.

No obvious solution: Creating an optimal algorithm to dynamically adjust traffic signals based on changing conditions is a non-trivial problem.

Addressing problems not encompassed by current standards and codes: Developing a system that goes beyond traditional traffic management standards and incorporates emerging technologies like autonomous vehicles.

Involving diverse groups of stakeholders: Collaboration with city authorities, transportation agencies, technology providers, and citizens is essential for successful implementation.

Many component parts or sub-problems: Designing algorithms for traffic prediction, data fusion, communication protocols, and adaptive control systems all contribute to the complexity.

Involving multiple disciplines: Requires expertise in computer science, data analytics, telecommunications, civil engineering, and urban planning.

Having significant consequences in a range of contexts: A well-designed traffic management system can significantly reduce congestion, improve air quality, enhance safety, and contribute to the overall efficiency of the city.

The development of a smart traffic management system exemplifies a problem that necessitates a complex digital solution due to its multifaceted nature and the integration of various technical and non-technical considerations.

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## When Your Go-To Problem-Solving Approach Fails

- Cheryl Strauss Einhorn

Eight steps to help you assess what’s not working — and why.

We make decisions all day, every day. The way we make decisions depends largely on context and our own unique problem-solving style. But, sometimes a tough workplace situation turns our usual problem-solving style on its head. Situationality is the culmination of many factors including location, life stage, decision ownership, and team dynamics. To make effective choices in the workplace, we often need to put our well-worn decision-making habits to the side and carefully ponder all aspects of the situation at hand.

Have you ever noticed that when you go home to your parents’ house, no matter what age you are, you make decisions differently than when you’re at work or out with a group of friends? For many of us, this is a familiar and sometimes frustrating experience — for example, allowing our parent to serve us more food than we want to eat. We feel like adults in our day-to-day lives, but when we step into our childhood homes we revert.

- Cheryl Strauss Einhorn is the founder and CEO of Decisive, a decision sciences company using her AREA Method decision-making system for individuals, companies, and nonprofits looking to solve complex problems. Decisive offers digital tools and in-person training, workshops, coaching and consulting. Cheryl is a long-time educator teaching at Columbia Business School and Cornell and has won several journalism awards for her investigative news stories. She’s authored two books on complex problem solving, Problem Solved for personal and professional decisions, and Investing In Financial Research about business, financial, and investment decisions. Her new book, Problem Solver, is about the psychology of personal decision-making and Problem Solver Profiles. For more information please watch Cheryl’s TED talk and visit areamethod.com .

## Partner Center

## Opinion Joe Biden: The U.S. won’t back down from the challenge of Putin and Hamas

Joe Biden is president of the United States.

Today, the world faces an inflection point, where the choices we make — including in the crises in Europe and the Middle East — will determine the direction of our future for generations to come.

What will our world look like on the other side of these conflicts?

Will we deny Hamas the ability to carry out pure, unadulterated evil? Will Israelis and Palestinians one day live side by side in peace, with two states for two peoples?

Will we hold Vladimir Putin accountable for his aggression, so the people of Ukraine can live free and Europe remains an anchor for global peace and security?

And the overarching question: Will we relentlessly pursue our positive vision for the future, or will we allow those who do not share our values to drag the world to a more dangerous and divided place?

Read this op-ed in Arabic.

Both Putin and Hamas are fighting to wipe a neighboring democracy off the map. And both Putin and Hamas hope to collapse broader regional stability and integration and take advantage of the ensuing disorder. America cannot, and will not, let that happen. For our own national security interests — and for the good of the entire world.

The United States is the essential nation. We rally allies and partners to stand up to aggressors and make progress toward a brighter, more peaceful future. The world looks to us to solve the problems of our time. That is the duty of leadership, and America will lead. For if we walk away from the challenges of today, the risk of conflict could spread, and the costs to address them will only rise. We will not let that happen.

That conviction is at the root of my approach to supporting the people of Ukraine as they continue to defend their freedom against Putin’s brutal war.

We know from two world wars in the past century that when aggression in Europe goes unanswered, the crisis does not burn itself out. It draws America in directly. That’s why our commitment to Ukraine today is an investment in our own security. It prevents a broader conflict tomorrow.

We are keeping American troops out of this war by supporting the brave Ukrainians defending their freedom and homeland. We are providing them with weapons and economic assistance to stop Putin’s drive for conquest, before the conflict spreads farther.

The United States is not doing this alone. More than 50 nations have joined us to ensure that Ukraine has what it needs to defend itself. Our partners are shouldering much of the economic responsibility for supporting Ukraine. We have also built a stronger and more united NATO , which enhances our security through the strength of our allies, while making clear that we will defend every inch of NATO territory to deter further Russian aggression. Our allies in Asia are standing with us as well to support Ukraine and hold Putin accountable, because they understand that stability in Europe and in the Indo-Pacific are inherently connected.

We have also seen throughout history how conflicts in the Middle East can unleash consequences around the globe.

We stand firmly with the Israeli people as they defend themselves against the murderous nihilism of Hamas. On Oct. 7, Hamas slaughtered 1,200 people, including 35 American citizens, in the worst atrocity committed against the Jewish people in a single day since the Holocaust. Infants and toddlers, mothers and fathers, grandparents, people with disabilities, even Holocaust survivors were maimed and murdered. Entire families were massacred in their homes . Young people were gunned down at a music festival. Bodies riddled with bullets and burned beyond recognition . And for over a month, the families of more than 200 hostages taken by Hamas, including babies and Americans, have been living in hell , anxiously waiting to discover whether their loved ones are alive or dead. At the time of this writing, my team and I are working hour by hour, doing everything we can to get the hostages released.

And while Israelis are still in shock and suffering the trauma of this attack , Hamas has promised that it will relentlessly try to repeat Oct. 7 . It has said very clearly that it will not stop.

The Palestinian people deserve a state of their own and a future free from Hamas. I, too, am heartbroken by the images out of Gaza and the deaths of many thousands of civilians, including children. Palestinian children are crying for lost parents. Parents are writing their child’s name on their hand or leg so they can be identified if the worst happens. Palestinian nurses and doctors are trying desperately to save every precious life they possibly can, with little to no resources. Every innocent Palestinian life lost is a tragedy that rips apart families and communities.

Our goal should not be simply to stop the war for today — it should be to end the war forever, break the cycle of unceasing violence , and build something stronger in Gaza and across the Middle East so that history does not keep repeating itself.

Just weeks before Oct. 7, I met in New York with Israeli Prime Minister Benjamin Netanyahu . The main subject of that conversation was a set of substantial commitments that would help both Israel and the Palestinian territories better integrate into the broader Middle East. That is also the idea behind the innovative economic corridor that will connect India to Europe through the United Arab Emirates, Saudi Arabia, Jordan and Israel, which I announced together with partners at the Group of 20 summit in India in early September. Stronger integration between countries creates predictable markets and draws greater investment. Better regional connection — including physical and economic infrastructure — supports higher employment and more opportunities for young people. That’s what we have been working to realize in the Middle East. It is a future that has no place for Hamas’s violence and hate, and I believe that attempting to destroy the hope for that future is one reason that Hamas instigated this crisis.

This much is clear: A two-state solution is the only way to ensure the long-term security of both the Israeli and Palestinian people. Though right now it may seem like that future has never been further away, this crisis has made it more imperative than ever.

A two-state solution — two peoples living side by side with equal measures of freedom, opportunity and dignity — is where the road to peace must lead. Reaching it will take commitments from Israelis and Palestinians, as well as from the United States and our allies and partners. That work must start now.

To that end, the United States has proposed basic principles for how to move forward from this crisis, to give the world a foundation on which to build.

To start, Gaza must never again be used as a platform for terrorism . There must be no forcible displacement of Palestinians from Gaza, no reoccupation, no siege or blockade, and no reduction in territory. And after this war is over, the voices of Palestinian people and their aspirations must be at the center of post-crisis governance in Gaza.

As we strive for peace, Gaza and the West Bank should be reunited under a single governance structure, ultimately under a revitalized Palestinian Authority, as we all work toward a two-state solution. I have been emphatic with Israel’s leaders that extremist violence against Palestinians in the West Bank must stop and that those committing the violence must be held accountable. The United States is prepared to take our own steps, including issuing visa bans against extremists attacking civilians in the West Bank.

The international community must commit resources to support the people of Gaza in the immediate aftermath of this crisis, including interim security measures, and establish a reconstruction mechanism to sustainably meet Gaza’s long-term needs. And it is imperative that no terrorist threats ever again emanate from Gaza or the West Bank.

If we can agree on these first steps, and take them together, we can begin to imagine a different future. In the months ahead, the United States will redouble our efforts to establish a more peaceful, integrated and prosperous Middle East — a region where a day like Oct. 7 is unthinkable.

In the meantime, we will continue working to prevent this conflict from spreading and escalating further. I ordered two U.S. carrier groups to the region to enhance deterrence. We are going after Hamas and those who finance and facilitate its terrorism, levying multiple rounds of sanctions to degrade Hamas’s financial structure, cutting it off from outside funding and blocking access to new funding channels, including via social media. I have also been clear that the United States will do what is necessary to defend U.S. troops and personnel stationed across the Middle East — and we have responded multiple times to the strikes against us.

I also immediately traveled to Israel — the first American president to do so during wartime — to show solidarity with the Israeli people and reaffirm to the world that the United States has Israel’s back. Israel must defend itself. That is its right. And while in Tel Aviv, I also counseled Israelis against letting their hurt and rage mislead them into making mistakes we ourselves have made in the past.

From the very beginning, my administration has called for respecting international humanitarian law, minimizing the loss of innocent lives and prioritizing the protection of civilians. Following Hamas’s attack on Israel, aid to Gaza was cut off, and food, water and medicine reserves dwindled rapidly. As part of my travel to Israel, I worked closely with the leaders of Israel and Egypt to reach an agreement to restart the delivery of essential humanitarian assistance to Gazans. Within days, trucks with supplies again began to cross the border. Today, nearly 100 aid trucks enter Gaza from Egypt each day, and we continue working to increase the flow of assistance manyfold. I’ve also advocated for humanitarian pauses in the conflict to permit civilians to depart areas of active fighting and to help ensure that aid reaches those in need. Israel took the additional step to create two humanitarian corridors and implement daily four-hour pauses in the fighting in northern Gaza to allow Palestinian civilians to flee to safer areas in the south.

This stands in stark opposition to Hamas’s terrorist strategy: hide among Palestinian civilians. Use children and innocents as human shields. Position terrorist tunnels beneath hospitals, schools, mosques and residential buildings. Maximize the death and suffering of innocent people — Israeli and Palestinian. If Hamas cared at all for Palestinian lives, it would release all the hostages, give up arms, and surrender the leaders and those responsible for Oct. 7.

As long as Hamas clings to its ideology of destruction, a cease-fire is not peace. To Hamas’s members, every cease-fire is time they exploit to rebuild their stockpile of rockets, reposition fighters and restart the killing by attacking innocents again. An outcome that leaves Hamas in control of Gaza would once more perpetuate its hate and deny Palestinian civilians the chance to build something better for themselves.

And here at home, in moments when fear and suspicion, anger and rage run hard, we have to work even harder to hold on to the values that make us who we are. We’re a nation of religious freedom and freedom of expression. We all have a right to debate and disagree and peacefully protest, but without fear of being targeted at schools or workplaces or elsewhere in our communities.

In recent years, too much hate has been given too much oxygen, fueling racism and an alarming rise in antisemitism in America. That has intensified in the wake of the Oct. 7 attacks. Jewish families worry about being targeted in school, while wearing symbols of their faith on the street or otherwise going about their daily lives. At the same time, too many Muslim Americans, Arab Americans and Palestinian Americans, and so many other communities, are outraged and hurting, fearing the resurgence of the Islamophobia and distrust we saw after 9/11.

We can’t stand by when hate rears its head. We must, without equivocation, denounce antisemitism, Islamophobia, and other forms of hate and bias. We must renounce violence and vitriol and see each other not as enemies but as fellow Americans.

In a moment of so much violence and suffering — in Ukraine, Israel, Gaza and so many other places — it can be difficult to imagine that something different is possible. But we must never forget the lesson learned time and again throughout our history: Out of great tragedy and upheaval, enormous progress can come. More hope. More freedom. Less rage. Less grievance. Less war. We must not lose our resolve to pursue those goals, because now is when clear vision, big ideas and political courage are needed most. That is the strategy that my administration will continue to lead — in the Middle East, Europe and around the globe. Every step we take toward that future is progress that makes the world safer and the United States of America more secure.

## About guest opinion submissions

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## IMAGES

## VIDEO

## COMMENTS

This is known as the complex version of the Fundamental Theorem of Calculus. Theorem 4.2.1. Let f(z) = F′ (z) be the derivative of a single-valued complex function F(z) defined on a domain Ω ⊂ C . Let C be any countour lying entirely in Ω with initial point z0 and final point z1.

This text constitutes a collection of problems for using as an additional learning resource for those who are taking an introductory course in complex analysis. The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex Numbers, Functions, Complex Integrals and Series. The majority of problems are ...

The usual properties of real line integrals are carried over to their complex counterparts. Some of these properties are: (i) Z C f(z) dz is independent of the parameterization of C; (ii) Z −C f(z) dz = − Z C f(z) dz, where −C is the opposite curve of C; (iii) The integrals of f(z) along a string of contours is equal to the

Complex integration Chapter 1 Complex integration 1.1 Complex number quiz Simplify 1 3+4i. Simplify | 1 3+4i|. Find the cube roots of 1. Here are some identities for complex conjugate. Which ones need correc- tion? z + w = z ̄ + w, ̄ z − w = z ̄ − w, ̄ zw = z ̄ w, ̄ z/w = z/ ̄ w.

Complex Integration Page 2 UNIT -IV COMPLEX INTEGRATION 4.1 LINE INTEGRAL AND CONTOUR INTEGRAL If ( V)is a continuous function of the complex variable V= T+𝑖 and C is any continuous curve connecting two points A and B on the z (- plane then the complex line integral of V) along C from A to B is denoted by ∫ ( V)

Complex Path Integrals. In this section we will investigate the computation of complex path integrals. Given two points in the complex plane, connected by a path \(\Gamma\) as shown in Figure \(\PageIndex{1}\), we would like to define the integral of \(f(z)\) along \(\Gamma\), \[\int_{\Gamma} f(z) d z\nonumber \] A natural procedure would be to work in real variables, by writing \[\int_{\Gamma ...

If you try calculating this using techniques that you know (integration by substitution, integration by parts, etc) then you will quickly hit an impasse. However, using complex analysis one can evaluate (1.1.1) in about ﬁve lines of work!1 §1.2 Recap on complex numbers A complex number is an expression of the form√ x+ iywhere x,y∈ R.

7.4: Exercises- Complex Integration. Let us confirm the representation of this Cauchy's Theorem equation in the matrix case. More precisely, if [Math Processing Error] Φ ( z) ≡ ( z I − B) − 1 is the transfer function associated with [Math Processing Error] B then this Cauchy's Theorem equation states that. Compute the [Math Processing ...

Complex Analysis: Problems with solutions Authors: Juan Carlos Ponce Campuzano The University of Queensland Abstract This text constitutes a collection of problems for using as an additional...

Everything about Cauchy's Integral Formula and examples on how to use it to solve complex integrals in complex analysis.Proof of the theorem can be found her...

5 Contour integrals 16 6 Cauchy's theorem 21 7 Consequences of Cauchy's theorem 26 8 Zeros, poles, and the residue theorem 35 ... seem like they ought to have little to do with complex numbers. For example: •Solving polynomial equations: historically, this was the motivation for ... •Solving physics problems in hydrodynamics, heat ...

Correct answer: πi 4 Explanation: Recall Cauchy's integral formula, which states f(z) = 1 2πi ∫C f(s) s − zds In this case, we have f(s) = cos(s) s2 + 8 Plugging in gives us ∫C cos z z2+8 z dz = 2πif(0) = 2πi1 8 = πi 4 Report an Error Example Question #1 : Complex Integration Let C be the unit circle. Compute ∫C z2sin(1 z) dz Possible Answers: −πi

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. ... Kindly help in solving the above problem. Thanks for giving time. integration; complex-analysis; complex-numbers; Share. Cite. Follow asked Dec 15, 2014 at 16:08. monalisa monalisa.

COMPLEX INTEGRATION | Problems and Solutions in Mathematics Major American Univ. Ph.D. Qualifying Questions and Solutions — Mathematics Problems and Solutions in Mathematics, pp. 377-412 (1998) No Access COMPLEX INTEGRATION https://doi.org/10.1142/9789812385406_0018 Cited by: 0 Previous Next PDF/EPUB Tools Share Recommend to Library Abstract:

They can often be derived using contour integrals. Sometimes we are indeed interested in contour integrals for complex parameters; quantum mechanics brought complex numbers to physics in an intrinsic way. This note introduces the contour integrals. 1 Basics of Contour Integrals Consider a two-dimensional plane (x,y), and regard it a "complex ...

Example 1: Integrate f (t) = 2t + it2. Here, u (t) = 2t and v (t) = t2. So, for 0≤t≤1. Solution: Example 2: Solve f (t) = iet. Integrating it from 0 to 2. Solution: Contour Integrals Contour integrals is basically the integration of the form, Where z 0 and z 1 are complex numbers. We immediately see a difficulty.

Here is a set of practice problems to accompany the Integration by Parts section of the Applications of Integrals chapter of the notes for Paul Dawkins Calculus II course at Lamar University. ... 1.7 Complex Numbers; 2. Solving Equations and Inequalities. 2.1 Solutions and Solution Sets; 2.2 Linear Equations;

There are various kinds of methods of solving complex integrations, such as using the Cauchy's integral theorem, the residue theorem or specifying a specific integration route. It is necessary to use the appropriate method for a complex integrations you want to solve. Of course, you should remember as many kinds of the methods as possible.

Many challenging integration problems can be solved surprisingly quickly by simply knowing the right technique to apply. While finding the right technique can be a matter of ingenuity, there are a dozen or so techniques that permit a more comprehensive approach to solving definite integrals.

Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. ... Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more. ... Enter a problem Cooking Calculators.

Learn about integrals using our free math solver with step-by-step solutions.

Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly ...

By Tessent Solutions. Intermittent bus and software deadlocks are amongst the toughest problems for development teams to detect and diagnose, particularly on complex SoCs with hundreds of cores, many shared resources and complex systems that include 2.5D and 3D integration. Hard-to-find corner cases can cause complex SoCs to hang or stall ...

if a given design exhibits one or more of these characteristics, it can be considered a complex engineering problem. 2.Complex Digital Solution Example: A problem that would require a complex digital solution is the development of a highly efficient and adaptive traffic management system for a smart city. Wide-ranging or conflicting technical ...

Summary. We make decisions all day, every day. The way we make decisions depends largely on context and our own unique problem-solving style. But, sometimes a tough workplace situation turns our ...

The world looks to us to solve the problems of our time. That is the duty of leadership, and America will lead. ... Stronger integration between countries creates predictable markets and draws ...