after a long searching for the analytic form of finite solenoid field, with the help of  from :

  1. Edmund E. Callaghan, S. H. (1960). The Magnetic Field of a Finite Solenoid (Techical note D-465). Washington, USA: Nation Aeronautics and Space Administration.
  2. Jackson, J. D. (1998). Classical Electrodynamics. John Wiley & Sons, Inc.
  3. Milton Abramowitz, I. A. (1965). Handbook of mathematical functions : with formulas, graphs, and mathematical tables. Dover.
  4. NIST Digital Library of Mathematical Functions. (n.d.). Retrieved from
we have the analytic form. Finite length Solenoid potential and field
the potential is:
A_\phi = \frac{\mu_0 I}{2\pi } \frac{1}{L} \sqrt{\frac{a}{\rho}} \left[ \zeta k \left( \frac{k^2+h^2-h^2k^2}{h^2k^2}K(k^2)-\frac{1}{k^2}E(k^2) +\frac{h^2-1}{h^2} \Pi(h^2,k^2) \right) \right]_{\zeta_-}^{\zeta_+}
the field vector is:
B_\rho = \frac{\mu_0 I}{2\pi} \frac{1}{L} \sqrt{\frac{a}{\rho}} \left[ \frac{k^2-2}{k}K(k^2) + \frac{2}{k} E(k^2)\right]_{\zeta_-}^{\zeta_+}
B_z =\frac{\mu_0 I}{2\pi} \frac{1}{L} \frac{1}{2 \sqrt{a \rho}} \left[ \zeta k \left(K(k^2) + \frac{a-\rho}{a+\rho} \Pi(h^2,k^2)\right)\right]_{\zeta_-}^{\zeta_+}
\zeta_{\pm}=z\pm \frac{L}{2}
K(m)=\int_0^{\pi/2}{\frac{1}{\sqrt{1-m sin^2 \theta }}} d\theta
E(m)=\int_0^{\pi/2}{\sqrt{1-m sin^2 \theta} } d\theta
\Pi(n,m)=\int_0^{\pi/2}{\frac{1}{(1-n sin^2 \theta)\sqrt{1-m sin^2 \theta }}} d\theta
those are complete elliptic integral of 1st , 2nd and third kind. i quoted them in here coz the argument is a bit confusing across different references and i settle it down on the context of this page. the below is the plotted field line and intensity for the coil dimension is 1 radius and 1 length.