Difference between revisions of "Q-Sin"

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(Properties)
(Properties)
 
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The function $\mathrm{Sin}_q$ is defined by
 
The function $\mathrm{Sin}_q$ is defined by
 
$$\mathrm{Sin}_q(z)=\dfrac{E_q(iz)-E_q(-iz)}{2i},$$
 
$$\mathrm{Sin}_q(z)=\dfrac{E_q(iz)-E_q(-iz)}{2i},$$
where $E_q$ denotes the [[q-exponential E|$q$-exponential $E$]].
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where $E_q$ denotes the [[q-exponential E sub q|$q$-exponential $E_q$]].
  
 
=Properties=
 
=Properties=
{{:q-Euler formula for E sub q}}
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[[q-Euler formula for E sub q]]<br />
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[[q-derivative of q-Sine]]<br />
<strong>Theorem:</strong> The following formula holds:
 
$$D_q \mathrm{Sin}_q(bz) = b \mathrm{Cos}_q(bz),$$
 
where $D_q$ is the [[q-difference operator]], $\mathrm{Sin}_q$ is the [[Q-Sin|$q$-Sine function]], and $\mathrm{Cos}_q$ is the [[Q-Cos|$q$-cosine function]].
 
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<strong>Proof:</strong> █
 
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=External links=
<strong>Theorem:</strong> The [[general solution]] of the [[q-difference equation|$q$-difference equation]] $D_q^2 y(x) + k^2 y(x) = 0$
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[http://homepage.tudelft.nl/11r49/documents/as98.pdf]
is $y(x)=c_1 \mathrm{Cos}_q(kz) + c_2 \mathrm{Sin}_q(kz).$
 
  
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=References=
<strong>Proof:</strong> █
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* {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=findme|next=q-Cos}}: (6.168)
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=References=
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[[Category:SpecialFunction]]
[http://homepage.tudelft.nl/11r49/documents/as98.pdf]
 

Latest revision as of 00:49, 15 September 2016

The function $\mathrm{Sin}_q$ is defined by $$\mathrm{Sin}_q(z)=\dfrac{E_q(iz)-E_q(-iz)}{2i},$$ where $E_q$ denotes the $q$-exponential $E_q$.

Properties

q-Euler formula for E sub q
q-derivative of q-Sine

External links

[1]

References