Difference between revisions of "Jacobi theta 1"

From specialfunctionswiki
Jump to: navigation, search
 
(11 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
__NOTOC__
 
__NOTOC__
 
Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_1$ function is defined by
 
Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_1$ function is defined by
$$\vartheta_1(z,q)=2q^{\frac{1}{4}} \displaystyle\sum_{k=0}^{\infty} (-1)^k q^{k(k+1)} \sin(2k+1)z,$$
+
$$\vartheta_1(z,q)=2q^{\frac{1}{4}} \displaystyle\sum_{k=0}^{\infty} (-1)^k q^{k(k+1)} \sin((2k+1)z),$$
 
where $\sin$ denotes the [[sine]] function.
 
where $\sin$ denotes the [[sine]] function.
 +
 +
<div align="center">
 +
<gallery>
 +
File:Jacobitheta1,q=0.5plot.png|Graph of $\vartheta_1(z,\frac{1}{2})$.
 +
File:Complexjacobitheta1,q=0.5plot.png|Domain coloring of $\vartheta_1 \left(z,\frac{1}{2} \right)$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
Line 8: Line 15:
 
[[Squares of theta relation for Jacobi theta 2 and Jacobi theta 4]]<br />
 
[[Squares of theta relation for Jacobi theta 2 and Jacobi theta 4]]<br />
 
[[Squares of theta relation for Jacobi theta 3 and Jacobi theta 4]]<br />
 
[[Squares of theta relation for Jacobi theta 3 and Jacobi theta 4]]<br />
[[Squares of theta relation for Jacobi theta 4 and Jacobi theta 4]]<br />
 
 
[[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 />
 +
[[Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines]]<br />
  
 
=References=
 
=References=
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Jacobi theta 2}}: 16.27.1
+
* {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=Jacobi theta 2}}: $164. (1)$
 +
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Jacobi theta 2}}: $16.27.1$
 +
 
 +
{{: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 $\vartheta_1$ function is defined by $$\vartheta_1(z,q)=2q^{\frac{1}{4}} \displaystyle\sum_{k=0}^{\infty} (-1)^k q^{k(k+1)} \sin((2k+1)z),$$ where $\sin$ denotes the sine 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
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 1 equals the log of a quotient of sines + a sum of sines

References

Jacobi theta functions