Difference between revisions of "Jacobi theta 4"

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$$\vartheta_4(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} (-1)^k q^{k^2} \cos(2kz),$$
 
$$\vartheta_4(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} (-1)^k q^{k^2} \cos(2kz),$$
 
where $\cos$ denotes the [[cosine]] function.
 
where $\cos$ denotes the [[cosine]] function.
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<div align="center">
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<gallery>
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File:Jacobitheta4,q=0.5plot.png|Graph of $\vartheta_4(z,\frac{1}{2})$.
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File:Complexjacobitheta4,q=0.5plot.png|Domain coloring of $\vartheta_4 \left(z,\frac{1}{2} \right)$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
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[[Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3]]<br />
 
[[Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3]]<br />
 
[[Derivative of Jacobi theta 1 at 0]]<br />
 
[[Derivative of Jacobi theta 1 at 0]]<br />
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[[Logarithm of a quotient of Jacobi theta 4 equals a sum of sines]]<br />
  
=See also=
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=References=
[[Jacobi theta 1]]<br />
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* {{BookReference|Special Functions|1960|Earl David Rainville|prev=Jacobi theta 3|next=findme}}: $164. (4)$
[[Jacobi theta 2]]<br />
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Jacobi theta 3|next=Squares of theta relation for Jacobi theta 1 and Jacobi theta 4}}: $16.27.4$
[[Jacobi theta 3]]<br />
 
  
=References=
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{{:Jacobi theta footer}}
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Jacobi theta 3|next=Squares of theta relation for Jacobi theta 1 and Jacobi theta 4}}: 16.27.4
 
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 20:24, 25 June 2017

Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_4$ function is defined by $$\vartheta_4(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} (-1)^k q^{k^2} \cos(2kz),$$ where $\cos$ denotes the cosine function.

Properties

Squares of theta relation for Jacobi theta 1 and Jacobi theta 4
Squares of theta relation for Jacobi theta 2 and Jacobi theta 4
Squares of theta relation for Jacobi theta 3 and Jacobi theta 4
Squares of theta relation for Jacobi theta 4 and Jacobi theta 4
Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3
Derivative of Jacobi theta 1 at 0
Logarithm of a quotient of Jacobi theta 4 equals a sum of sines

References

Jacobi theta functions