Jacobi zeta function

In mathematics, the Jacobi zeta function Z(u) is the <a href="/facts/Logarithmic_derivative/VL7GIwkl">logarithmic derivative</a> of the <a href="/facts/Jacobi_theta_function/g09xvoQQ">Jacobi theta function</a> Θ(u). It is also commonly denoted as 
 
 
 
 zn
 ⁡
 (
 u
 ,
 k
 )
 
 
 {\displaystyle \operatorname {zn} (u,k)}

Θ
        (
        u
        )
        =
        
          Θ
          
            4
          
        
        
          (
          
            
              
                π
                u
              
              
                2
                K
              
            
          
          )
        
      
    
    {\displaystyle \Theta (u)=\Theta _{4}\left({\frac {\pi u}{2K}}\right)}

Z
 (
 u
 )
 =
 
 
 ∂
 
 ∂
 u
 
 
 
 ln
 ⁡
 Θ
 (
 u
 )
 
 
 {\displaystyle Z(u)={\frac {\partial }{\partial u}}\ln \Theta (u)}
 
 
 
 
 
 =
 
 
 
 
 Θ
 ′
 
 (
 u
 )
 
 
 Θ
 (
 u
 )
 
 
 
 
 
 {\displaystyle ={\frac {\Theta '(u)}{\Theta (u)}}}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

Z
 (
 ϕ
 
 |
 
 m
 )
 =
 E
 (
 ϕ
 
 |
 
 m
 )
 −
 
 
 
 E
 (
 m
 )
 
 
 K
 (
 m
 )
 
 
 
 F
 (
 ϕ
 
 |
 
 m
 )
 
 
 {\displaystyle Z(\phi |m)=E(\phi |m)-{\frac {E(m)}{K(m)}}F(\phi |m)}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Where E, K, and F are generic Incomplete <a href="/facts/Elliptic_integral/q8EGYzQ3">Elliptical Integrals</a> of the first and second kind. Jacobi Zeta Functions being kinds of Jacobi theta functions have applications to all their relevant fields and application.

zn
 ⁡
 (
 u
 ,
 k
 )
 =
 Z
 (
 u
 )
 =
 
 ∫
 
 0
 
 
 u
 
 
 
 dn
 
 2
 
 
 ⁡
 v
 −
 
 
 E
 K
 
 
 d
 v
 
 
 {\displaystyle \operatorname {zn} (u,k)=Z(u)=\int _{0}^{u}\operatorname {dn} ^{2}v-{\frac {E}{K}}dv}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
This relates Jacobi's common notation of, 
 
 
 
 dn
 ⁡
 
 u
 
 =
 
 
 1
 −
 m
 sin
 ⁡
 
 
 θ
 
 
 2
 
 
 
 
 
 
 {\displaystyle \operatorname {dn} {u}={\sqrt {1-m\sin {\theta }^{2}}}}
 
, 
 
 
 
 sn
 ⁡
 u
 =
 sin
 ⁡
 
 θ
 
 
 
 {\displaystyle \operatorname {sn} u=\sin {\theta }}
 
, 
 
 
 
 cn
 ⁡
 u
 =
 cos
 ⁡
 
 θ
 
 
 
 {\displaystyle \operatorname {cn} u=\cos {\theta }}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> to Jacobi's Zeta function.
Some additional relations include ,

zn
 ⁡
 (
 u
 ,
 k
 )
 =
 
 
 π
 
 2
 K
 
 
 
 
 
 
 
 Θ
 
 1
 
 ′
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 Θ
 
 1
 
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 −
 
 
 
 cn
 ⁡
 
 u
 
 
 dn
 ⁡
 
 u
 
 
 
 sn
 ⁡
 
 u
 
 
 
 
 
 
 {\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{1}'{\frac {\pi u}{2K}}}{\Theta _{1}{\frac {\pi u}{2K}}}}-{\frac {\operatorname {cn} {u}\,\operatorname {dn} {u}}{\operatorname {sn} {u}}}}
 
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

zn
 ⁡
 (
 u
 ,
 k
 )
 =
 
 
 π
 
 2
 K
 
 
 
 
 
 
 
 Θ
 
 2
 
 ′
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 Θ
 
 2
 
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 −
 
 
 
 sn
 ⁡
 
 u
 
 
 dn
 ⁡
 
 u
 
 
 
 cn
 ⁡
 
 u
 
 
 
 
 
 
 {\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{2}'{\frac {\pi u}{2K}}}{\Theta _{2}{\frac {\pi u}{2K}}}}-{\frac {\operatorname {sn} {u}\,\operatorname {dn} {u}}{\operatorname {cn} {u}}}}
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

zn
 ⁡
 (
 u
 ,
 k
 )
 =
 
 
 π
 
 2
 K
 
 
 
 
 
 
 
 Θ
 
 3
 
 ′
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 Θ
 
 3
 
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 −
 
 k
 
 2
 
 
 
 
 
 sn
 ⁡
 
 u
 
 
 cn
 ⁡
 
 u
 
 
 
 dn
 ⁡
 
 u
 
 
 
 
 
 
 {\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{3}'{\frac {\pi u}{2K}}}{\Theta _{3}{\frac {\pi u}{2K}}}}-k^{2}{\frac {\operatorname {sn} {u}\,\operatorname {cn} {u}}{\operatorname {dn} {u}}}}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

zn
 ⁡
 (
 u
 ,
 k
 )
 =
 
 
 π
 
 2
 K
 
 
 
 
 
 
 
 Θ
 
 4
 
 ′
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 Θ
 
 4
 
 
 
 
 
 π
 u
 
 
 2
 K
 
 
 
 
 
 
 
 
 {\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{4}'{\frac {\pi u}{2K}}}{\Theta _{4}{\frac {\pi u}{2K}}}}}
 
<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<ul><li><a href="https://booksite.elsevier.com/samplechapters/9780123736376/Sample_Chapters/01~Front_Matter.pdf">https://booksite.elsevier.com/samplechapters/9780123736376/Sample_Chapters/01~Front_Matter.pdf</a> Pg.xxxiv</li>
<li><a href="/facts/Milton_Abramowitz/6yRyez7P">Abramowitz, Milton</a>; <a href="/facts/Irene_Stegun/FjSS1LDR">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a href="http://www.math.ubc.ca/~cbm/aands/page_578.htm">"Chapter 16"</a>. <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 578. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61272-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/65012253">65-12253</a>.</li>
<li><a href="http://mathworld.wolfram.com/JacobiZetaFunction.html">http://mathworld.wolfram.com/JacobiZetaFunction.html</a></li></ul>

Jacobi zeta function open-in-new

Jacobi zeta function