• Canonical solutions of Bessel's differential equation: \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx} + (x^2-\alpha^2)y = 0. \]
  • Regular Singularity: \(0\)
  • Irregular Singularity: \(\infty\)
• Various formulation:
  • Bessel Functions
  • Modified Bessel Functions
  • Hankel Functions
  • Spherical Bessel Functions
  • Spherical Hankel Functions

• \[ J_\alpha(x) =\sum_{m=0}^\infty \frac{(-1)^m}{m!\Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha} \]
  • This is obtained by the ./Differential Equation.html#org378627a.

• Weber Functions, Neumann Functions
• \(Y_\alpha(x), N_\alpha(x)\)
• \[ Y_\alpha(x) = \frac{J_\alpha(x)\cos(\alpha\pi)-J_{-\alpha}(x)}{\sin(\alpha\pi)} \]
• In the case of integer order \(n\), the function is defined by the limit:
  • \[ Y_n(x) = \lim_{\alpha\to n}Y_\alpha(x) \]

• Bessel functions for half-integer \(\alpha\)
• The radial part of the solution to ./Differential Equation.html#org7493b51 in spherical coordinates.

• Linearly independent functions that are solutions to ./Differential Equation.html#orgfb166e0 in cylindrical coordinates.
• Each basis is the product of three functions \[ V_n(k) = R_n(k, \rho)\Phi_n(\varphi)Z(k, z). \]
• \(\Rho\)\( R_n(k, \rho) \) is given by the 5.

• Gaussian Hypergeometric Function
• \(_2F_1(a, b; c; z)\)

• Confluent Hypergeometric Function of the First Kind
• \[ M(a,b,z) = {}_1F_1(a;b;z) := \sum_{n=0}^\infty \frac{a^{\overline{n}}}{b^{\overline{n}}}\frac{z^n}{n!} \]
• Unrelated another Kummer's function for the record.
  • \[ \Lambda_n(z) := \int_0^z \frac{\log^{n-1}|t|}{1+t}\,dt \]

• \[ U(a,b,z) := \frac{\Gamma(1-b)}{\Gamma(a+1-b)}M(a,b,z) + \frac{\Gamma(b-1)}{\Gamma(a)} z^{1-b}M(a+1-b, 2-b, z). \]
• Another form of the solution to the Kummer's differential equation

• Generalized Hypergeometric Series
• Convergent power series in which the ratio of successive coefficients indexed by \(n\) is a rational function of \(n\).
• Formally, a hypergeometric series is a power series:
  • \[ \sum_{n\ge 0} \beta_nz^n \]
  • with
    • \[ \forall n\ge 0, \frac{\beta_{n+1}}{\beta_n} = \frac{A(n)}{B(n)} \]
    • where \(A(n), B(n)\) being polynomials in \(n\).
• Customarily, The leading term \(\beta_0\) is factored out, and \(B(n)\) is assumed to have the factor \((1+n)\). This is without loss of generality, since both \(A\) and \(B\) can be multiplied by \((1+n)\) if needed.
• Then using the linear factors over the complex numbers and scaling \(z\):
  • \[ {}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q; z) = {}_pF_q\left[\begin{array}{cccc}a_1 & a_2 & \cdots & a_p \\ b_1 & b_2 & \cdots & b_q \end{array}; z\right] = \sum_{n=0}^\infty\frac{a_1^{\overline{n}}\cdots a_p^{\overline{n}}}{b_1^{\overline{n}}\cdots b_q^{\overline{n}}}\frac{z^n}{n!}. \]
  • It is in the form of exponential generating function.

• Function \(f\) which can be expressed in the form: \[ f(x) = \int_c^x R(t, \sqrt{P(t)})\,dt \] where \(R\) is a rational function, and \(P\) is a polynomial of degree 3 or 4 with no repeated roots.

• By convention, the names of the arguments are:
  • \(k\): Elliptic Modulus or ../geometry/Conic Sections.html#org08c6736.
  • \(\alpha\): Modular Angle, \(\sin\alpha = k\)
  • \(m\): Parameter, \(m = k^2\)

• It is the inverse of the 8.4.1.
  • \[ \operatorname{am}(u, m) = \varphi = F_m^{-1}(u) : F(\varphi, m) = \int_0^\varphi \frac{d\theta}{\sqrt{1-m\sin^2\theta}} = u. \]
    • It is equal to the angle when \(m=0\), and increases as the \(m\) increases.

#+begin_html q c s n d p c (x) 1 x/y=cot(φ) x/r=cos(φ) x=cos(φ)/dn s (y) y/x=tan(φ) 1 y/r=sin(φ) y=sin(φ)/dn n (r) r/x=sec(φ) r/y=csc(φ) 1 r=1/dn d (1) 1/x=sec(φ)dn 1/y=csc(φ)dn 1/r=dn 1 #+end_html -

• \[ \rm pq\cdot p'q' = pq'\cdot p'q \] which implies:
  • \[ \rm \frac{pr}{qr} = pq \]
  • \[ \rm pr\cdot rq = pq \]
  • \[ \rm \frac{1}{qp} = pq \]

• \(\operatorname{cn}(u)^2 + \operatorname{sn}(u)^2 = 1\)
• \(\operatorname{dn}(u)^2 + m\operatorname{sn}(u)^2 = 1\)
  • See the property of ellipse.
• \(\operatorname{dn}(u)^2 + (1-m)\operatorname{sn}(u)^2 = \operatorname{cn}(u)^2\)

• \[ \phi(q) = \prod_{k=1}^\infty (1-q^k) \] with \(|q| < 1\).

• Euler Identity
• \[ \phi(q) = \sum_{n=-\infty}^\infty (-1)^n q^{(3n^2 - n)/2}. \]
• \((3n^2 - n)/2\) is a pentagonal number.

• The inverse of \[ f(w) = we^w = z. \]
• \(W(z) = f^{-1}(z)\).
• \(W_0(z)\) is the principal branch.