Synchrotron function

File:Morera's Theorem.png

If the integral along every C is zero, then ƒ is holomorphic on D.

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

Morera's theorem states that a continuous, complex-valued function ƒ defined on a simply connected open set D in the complex plane that satisfies

${\oint }_{\gamma }f(z)dz=0$

for every closed piecewise C¹ curve $\gamma$ in D must be holomorphic on D.

The assumption of Morera's theorem is equivalent to that ƒ has an antiderivative on D.

The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero.

Proof

File:Morera's Theorem Proof.png

The integrals along two paths from a to b are equal, since their difference is the integral along a closed loop.

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for ƒ explicitly. The theorem then follows from the fact that holomorphic functions are analytic.

Without loss of generality, it can be assumed that D is connected. Fix a point z₀ in D, and for any $z\in D$, let $\gamma :[0,1]\to D$ be a piecewise C¹ curve such that $\gamma (0)=z_0$ and $\gamma (1)=z$. Then define the function F to be

$F(z)=\underset{\gamma }{\int }f(\zeta )d\zeta .$

To see that the function is well-defined, suppose $\tau :[0,1]\to D$ is another piecewise C¹ curve such that $\tau (0)=z_0$ and $\tau (1)=z$. The curve $\gamma {\tau }^{-1}$ (i.e. the curve combining $\gamma$ with $\tau$ in reverse) is a closed piecewise C¹ curve in D. Then,

${\oint }_{\gamma }f(\zeta )d\zeta +{\oint }_{{\tau }^{-1}}f(\zeta )d\zeta ={\oint }_{\gamma {\tau }^{-1}}f(\zeta )d\zeta =0$

And it follows that

${\oint }_{\gamma }f(\zeta )d\zeta ={\oint }_{\tau }f(\zeta )d\zeta .$

By continuity of f and the definition of the derivative, we get that F'(z) = f(z). Note that we can apply neither the Fundamental theorem of Calculus nor the mean value theorem since they are only true for real-valued functions.

Since f is the derivative of the holomorphic function F, it is holomorphic. This completes the proof.

Applications

Morera's theorem is a standard tool in complex analysis. It is used in almost any argument that involves a non-algebraic construction of a holomorphic function.

Uniform limits

For example, suppose that ƒ₁, ƒ₂, ... is a sequence of holomorphic functions, converging uniformly to a continuous function ƒ on an open disc. By Cauchy's theorem, we know that

${\oint }_C{f}_n(z)dz=0$

for every n, along any closed curve C in the disc. Then the uniform convergence implies that

${\oint }_{C}f(z)dz={\oint }_C\underset{n\to \mathrm{\infty }}{\lim }{f}_n(z)dz=\underset{n\to \mathrm{\infty }}{\lim }{\oint }_C{f}_n(z)dz=0$

for every closed curve C, and therefore by Morera's theorem ƒ must be holomorphic. This fact can be used to show that, for any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a Banach space with respect to the supremum norm.

Infinite sums and integrals

Morera's theorem can also be used in conjunction with Fubini's theorem and the Weierstrass M-test to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function

$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$

or the Gamma function

$\mathrm{\Gamma }(\alpha )=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{\alpha -1}{e}^{-x}dx.$

Specifically one shows that

${\oint }_C\mathrm{\Gamma }(\alpha )d\alpha =0$

for a suitable closed curve C, by writing

${\oint }_C\mathrm{\Gamma }(\alpha )d\alpha ={\oint }_C\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{\alpha -1}{e}^{-x}dxd\alpha$

and then using Fubini's theorem to justify changing the order of integration, getting

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{\oint }_C{x}^{\alpha -1}{e}^{-x}d\alpha dx=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}{\oint }_C{x}^{\alpha -1}d\alpha dx.$

Then one uses the analyticity of x ↦ x^α−1 to conclude that

${\oint }_C{x}^{\alpha -1}d\alpha =0,$

and hence the double integral above is 0. Similarly, in the case of the zeta function, the M-test justifies interchanging the integral along the closed curve and the sum.

Weakening of hypotheses

The hypotheses of Morera's theorem can be weakened considerably. In particular, it suffices for the integral

${\oint }_{\partial T}f(z)dz$

to be zero for every closed triangle T contained in the region D. This in fact characterizes holomorphy, i.e. ƒ is holomorphic on D if and only if the above conditions hold.

See also

• Cauchy–Riemann equations
• Methods of contour integration
• Residue (complex analysis)
• Mittag-Leffler's theorem

References

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  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.
  
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  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.
  
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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
• 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.
• 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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• Module for Morera's Theorem by John H. Mathews
• EoM article