Difference between revisions of "Linear fraction"
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$(1) ~ ~ ~ \displaystyle T(z)=\frac{u+v z}{w+z}$ |
$(1) ~ ~ ~ \displaystyle T(z)=\frac{u+v z}{w+z}$ |
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| + | |||
| + | where $u$, $v$, $w$ are parameters from the some set of numbers that allows operations of summation, multiplication and division. |
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| + | Usually, it is assumed, that they are complex numbers, and the operation of multiplication is commutative. |
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==Linear function== |
==Linear function== |
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$(2) ~ ~ ~ \displaystyle A+B z= \lim_{M\rightarrow \infty} \frac{M A+ M B}{M+z}$ |
$(2) ~ ~ ~ \displaystyle A+B z= \lim_{M\rightarrow \infty} \frac{M A+ M B}{M+z}$ |
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| − | where the expression of the function under the limit operation is expressed in a form that corresponds to (1) |
+ | where the expression of the function under the limit operation is expressed in a form that corresponds to (1), id est, |
| + | $u=M A$, $v=MB$, $w=M$. |
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==Inverse function== |
==Inverse function== |
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| + | The inverse function $T^{-1}$ of the linear fraction $T$ by (1) is also linear fraction, and its parameters can be easy expressed through the parameters of the initial linear fraction. |
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| + | |||
| + | $(3) ~ ~ ~ \displaystyle T^{-1}(z)=\frac{u-w z}{-v+z}$ |
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| + | |||
| + | One can easy check that $T(T^{-1}(z))=T^{-1}(T(z))=z$ |
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| + | for all $z$ excluding the poles, singularities at $z=-w$ and at $z=v$. |
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==Keywords== |
==Keywords== |
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Revision as of 14:23, 31 August 2013
Linear fraction is meromorphic function that can be expressed with
$(1) ~ ~ ~ \displaystyle T(z)=\frac{u+v z}{w+z}$
where $u$, $v$, $w$ are parameters from the some set of numbers that allows operations of summation, multiplication and division. Usually, it is assumed, that they are complex numbers, and the operation of multiplication is commutative.
Linear function
Definition (1) excludes the case of linear function. However, this the linear function can be realized in limit
$(2) ~ ~ ~ \displaystyle A+B z= \lim_{M\rightarrow \infty} \frac{M A+ M B}{M+z}$
where the expression of the function under the limit operation is expressed in a form that corresponds to (1), id est, $u=M A$, $v=MB$, $w=M$.
Inverse function
The inverse function $T^{-1}$ of the linear fraction $T$ by (1) is also linear fraction, and its parameters can be easy expressed through the parameters of the initial linear fraction.
$(3) ~ ~ ~ \displaystyle T^{-1}(z)=\frac{u-w z}{-v+z}$
One can easy check that $T(T^{-1}(z))=T^{-1}(T(z))=z$ for all $z$ excluding the poles, singularities at $z=-w$ and at $z=v$.
