Disjunctive normal form

In complex analysis, a field in mathematics, the residue theorem, sometimes called Cauchy's residue theorem (one of many things named after Augustin-Louis Cauchy), is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. 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.

Illustration of the setting.

The statement is as follows:

Suppose U is a simply connected open subset of the complex plane, and a₁,...,a_n are finitely many points of U and f is a function which is defined and holomorphic on U \ {a₁,...,a_n}. If γ is a rectifiable curve in U which does not meet any of the a_k, and whose start point equals its endpoint, then

$\oint _{\gamma }f(z)\,dz=2\pi i\sum _{k=1}^{n}\operatorname {I} (\gamma ,a_{k})\operatorname {Res} (f,a_{k}).$

If γ is a positively oriented simple closed curve, I(γ, a_k) = 1 if a_k is in the interior of γ, and 0 if not, so

$\oint _{\gamma }f(z)\,dz=2\pi i\sum \operatorname {Res} (f,a_{k})$

with the sum over those k for which a_k is inside γ.

Here, Res(f, a_k) denotes the residue of f at a_k, and I(γ, a_k) is the winding number of the curve γ about the point a_k. This winding number is an integer which intuitively measures how many times the curve γ winds around the point a_k; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around a_k and 0 if γ doesn't move around a_k at all.

The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve γ must first be reduced to a set of simple closed curves {γ_i} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of f dz along a Jordan curve γ_i with interior V. The requirement that f be holomorphic on U₀ = U \ {a_k} is equivalent to the statement that the exterior derivative d(f dz) = 0 on U₀. Thus if two planar regions V and W of U enclose the same subset {a_j} of {a_k}, the regions V\W and W\V lie entirely in U₀, and hence $\scriptstyle \int _{V\backslash W}d(f\,dz)-\int _{W\backslash V}d(f\,dz)$ is well-defined and equal to zero. Consequently, the contour integral of f dz along γ_j = ∂V is equal to the sum of a set of integrals along paths λ_j, each enclosing an arbitrarily small region around a single a_j—the residues of f (up to the conventional factor 2πi) at {a_j}. Summing over {γ_j}, we recover the final expression of the contour integral in terms of the winding numbers {I(γ, a_k)}.

In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.

Example

The integral

$\int _{-\infty }^{\infty }{e^{itx} \over x^{2}+1}\,dx$

The contour C.

arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals.

Suppose t > 0 and define the contour C that goes along the real line from −a to a and then counterclockwise along a semicircle centered at 0 from a to −a. Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. The contour integral is

$\int _{C}{f(z)}\,dz=\int _{C}{e^{itz} \over z^{2}+1}\,dz.$

Since e^itz is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator z² + 1 is zero. Since z² + 1 = (z + i)(z − i), that happens only where z = i or z = −i. Only one of those points is in the region bounded by this contour. Because f(z) is

${\begin{aligned}{\frac {e^{itz}}{z^{2}+1}}&={\frac {e^{itz}}{2i}}\left({\frac {1}{z-i}}-{\frac {1}{z+i}}\right)\\&={\frac {e^{itz}}{2i(z-i)}}-{\frac {e^{itz}}{2i(z+i)}},\end{aligned}}$

the residue of f(z) at z = i is

$\operatorname {Res} \limits _{z=i}f(z)={e^{-t} \over 2i}.$

According to the residue theorem, then, we have

$\int _{C}f(z)\,dz=2\pi i\cdot \operatorname {Res} \limits _{z=i}f(z)=2\pi i{e^{-t} \over 2i}=\pi e^{-t}.$

The contour C may be split into a "straight" part and a curved arc, so that

$\int _{\mathrm {straight} }f(z)\,dz+\int _{\mathrm {arc} }f(z)\,dz=\pi e^{-t}\,$

and thus

$\int _{-a}^{a}f(z)\,dz=\pi e^{-t}-\int _{\mathrm {arc} }f(z)\,dz.$

Using some estimations, we have

$\left|\int _{\mathrm {arc} }{e^{itz} \over z^{2}+1}\,dz\right|\leq \int _{\mathrm {arc} }\left|{e^{itz} \over z^{2}+1}\right|dz\leq \int _{\mathrm {arc} }{1 \over |z^{2}+1|}dz\leq \int _{\mathrm {arc} }{1 \over a^{2}-1}dz={\frac {\pi a}{a^{2}-1}}.$

and

$\lim _{a\to \infty }{\frac {\pi a}{a^{2}-1}}=0.$

Note that, since t > 0 and for complex numbers in the upper halfplane the argument lies between 0 and π, one can estimate

$\left|e^{itz}\right|=\left|e^{it|z|(\cos \phi +i\sin \phi )}\right|=\left|e^{-t|z|\sin \phi +it|z|\cos \phi }\right|=e^{-t|z|\sin \phi }\leq 1.$

Therefore

$\int _{-\infty }^{\infty }{e^{itz} \over z^{2}+1}\,dz=\pi e^{-t}.$

If t < 0 then a similar argument with an arc C' that winds around −i rather than i shows that

The contour C'.

$\int _{-\infty }^{\infty }{e^{itz} \over z^{2}+1}\,dz=\pi e^{t},$

and finally we have

$\int _{-\infty }^{\infty }{e^{itz} \over z^{2}+1}\,dz=\pi e^{-\left|t\right|}.$

(If t = 0 then the integral yields immediately to elementary calculus methods and its value is π.)

See also

• Cauchy's integral formula
• Jordan's lemma
• Methods of contour integration
• Morera's theorem
• Nachbin's theorem
• Residue at infinity
• Logarithmic form

References

General references

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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• Residue theorem in MathWorld
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