Difference between revisions of "Jacobi theta 2"

From specialfunctionswiki
Jump to: navigation, search
 
(2 intermediate revisions by the same user not shown)
Line 3: Line 3:
 
$$\vartheta_2(z,q)=2q^{\frac{1}{4}}\displaystyle\sum_{k=0}^{\infty} q^{k(k+1)} \cos(2k+1)z,$$
 
$$\vartheta_2(z,q)=2q^{\frac{1}{4}}\displaystyle\sum_{k=0}^{\infty} q^{k(k+1)} \cos(2k+1)z,$$
 
where $\cos$ denotes the [[cosine]] function.
 
where $\cos$ denotes the [[cosine]] function.
 +
 +
<div align="center">
 +
<gallery>
 +
File:Jacobitheta2,q=0.5plot.png|Graph of $\vartheta_2(z,\frac{1}{2})$.
 +
File:Complexjacobitheta2,q=0.5plot.png|Domain coloring of $\vartheta_2 \left(z,\frac{1}{2} \right)$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
Line 12: Line 19:
 
[[Derivative of Jacobi theta 1 at 0]]<br />
 
[[Derivative of Jacobi theta 1 at 0]]<br />
 
[[Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines]]<br />
 
[[Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines]]<br />
 
=See also=
 
[[Jacobi theta 1]]<br />
 
[[Jacobi theta 3]]<br />
 
[[Jacobi theta 4]]<br />
 
  
 
=References=
 
=References=
 
* {{BookReference|Special Functions|1960|Earl David Rainville|prev=Jacobi theta 1|next=Jacobi theta 3}}: $164. (2)$
 
* {{BookReference|Special Functions|1960|Earl David Rainville|prev=Jacobi theta 1|next=Jacobi theta 3}}: $164. (2)$
 
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Jacobi theta 1|next=Jacobi theta 3}}: $16.27.2$
 
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Jacobi theta 1|next=Jacobi theta 3}}: $16.27.2$
 +
 +
{{:Jacobi theta footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 20:23, 25 June 2017

Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_2$ function is defined by $$\vartheta_2(z,q)=2q^{\frac{1}{4}}\displaystyle\sum_{k=0}^{\infty} q^{k(k+1)} \cos(2k+1)z,$$ 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 quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines

References

Jacobi theta functions