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Confluent hypergeometric functions solve the Kummer differential equation $z{y}^{\prime \prime }+(b-z)y^{\prime }-ay=0$. The regular solution is the confluent hypergeometric function of the first kind; a second standard solution is conventionally denoted $U(a,b,z)$. The family arises by coalescing singular points of the Gauss hypergeometric equation.

For $b$ not a nonpositive integer, the regular solution normalized by $M(a,b,0)=1$ has the power seriesHere $(a{)}_n$ is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.

The origin is a regular singular point of this Kummer differential equation. Apply the Frobenius method, writing $y={\sum }_{n=0}^{\infty }{a}_n{x}^{n+r}$ with $a_0\ne 0$. The lowest power gives the indicial equationFor $n\ge 1$, comparison of the coefficient of $x^{n+r-1}$ givesSince $0<c<1$, both roots give valid independent branches. Normalize $a_0=1$:Here $(c{)}_n$ is the rising factorial, and the first series is the confluent hypergeometric function of the first kind $M(1,c,x)$. The two leading powers are distinct, proving linear independence. We describe real solutions on $x>0$, with their one-sided behavior at zero.

For the inhomogeneous equation a particular solution is $-(x+c)/2$, verified by direct substitution. Thus $y=A{y}_1+B{y}_2-(x+c)/2$. Since $y_2^{\prime }\sim (1-c)x^{-c}$, a finite derivative at zero forces $B=0$. Then $y(0)=0$ forces $A=c/2$, and the required solution isIn particular $y^{\prime }(0)=0$. The singular homogeneous branch cannot be added without violating the derivative condition.

Let $\eta >0$ denote the uniform magnetic diffusivity. The resistive induction equation for a solenoidal vector field in an incompressible flow isHere the radial compression and axial stretching have zero total divergence: $r^{-1}{\partial }_r(-\omega r^2)+{\partial }_z(2\omega z)=0$. A purely axial field independent of $z$ is automatically solenoidal, and the stretching term is $2\omega B_z\hat{\mathbf{z}}$. Thus the ansatz closes under the equation. For its angular mode, with $m$ a nonnegative integer,This is magnetic concentration by an incompressible stagnation flow.

Put $q=r/g(t)$ and $D_{m}b=b^{\prime \prime }+q^{-1}{b}^{\prime }-m^2{q}^{-2}b$. Substitution and division by $f$ giveA convenient self-similar magnetic mode in a stagnation flow is obtained by making the two dimensionless separation coefficients constant. ChooseThe constant $\lambda$ labels the radial modes. More generally the first right side can be $\beta \eta$ and the radial drift term $\beta q{b}^{\prime }$; the displayed equations choose the positive-width normalization $\beta =1$.

If $g(0)=g_0>0$, the width isFor $\omega =0$ its continuous limiting expression is $g^2=g_0^2+2\eta t$. Since $\eta /g^2=\omega +g^{\prime }/g$, the amplitude isFor the compressive case $\omega >0$, the width tends to the diffusion-strain balance scale $\sqrt{\eta /\omega }$ rather than collapsing to zero.

Let $I_b={\int }_0^{\infty }b(q{)}^{2}qdq$ be finite, and let $C_m={\int }_0^{2\pi }{\cos }^2(m\varphi )d\varphi$, equal to $2\pi$ for $m=0$ and $\pi$ otherwise. The magnetic energy per unit axial length isso the complete time dependence isThe angular integral is only a constant; zero signed magnetic flux does not force $I_b$ or $E$ to vanish.

For $m=0$, smoothness on the axis requires $b(0)$ finite and $b^{\prime }(0)=0$. Set $s=-q^2/2$ and $b=h(s)$. The radial equation becomes the Kummer differential equationIts regular solution iswhere $M$ is the confluent hypergeometric function of the first kind. The second independent solution has a logarithmic singularity at the axis and is inadmissible for a smooth field. For real $\lambda >0$, the regular solution vanishes at infinity: generically it has a $q^{-\lambda }$ tail; at positive even integer $\lambda$ it instead has a Gaussian times a polynomial. Thus decay at infinity alone does not select a unique separation parameter. Generically finite magnetic energy further requires $\lambda >1$.

The simplest finite-flux solution is $\lambda =2$, for which $M(1,1,s)=e^s$ andDirect differentiation gives $b^{\prime \prime }+q^{-1}{b}^{\prime }=(q^2-2)b$ and $q{b}^{\prime }=-q^{2}b$, verifying the radial equation. Its axial magnetic flux is $2\pi b_{0}f{g}^2=2\pi b_0{f}_0{g}_0^2$, while its magnetic energy grows by $g_0^2/g^2$ if the original width is larger than $\sqrt{\eta /\omega }$. For positive $\eta$ this Gaussian amplification saturates, rather than giving indefinite exponential growth. The other rapidly decaying regular axisymmetric modes have $\lambda =2(n+1)$ and profiles $e^{-q^2/2}{L}_n(q^2/2)$, where $L_n$ solves $y{L}_n^{\prime \prime }+(1-y)L_n^{\prime }+n{L}_n=0$.

The energy mechanism is transparent from a separate integral calculation. For any sufficiently decaying field of this axial form, integration by parts in the transverse plane givesThe positive term represents work by the axial stretching, with radial compression accounted for in the area integral. In the ideal limit, $g=g_0{e}^{-\omega t}$ and $f=f_0{e}^{2\omega t}$ for any smooth profile, so $E=E_0{e}^{2\omega t}$ even when $m\ne 0$ and the angular mean is zero. For sufficiently weak initial diffusion, this also gives genuine finite-time energy amplification at positive $\eta$.

Even sustained energy growth is possible in this unbounded model if slowly decaying profiles are admitted. For example, with $\omega >0$, take the constant width $g=\sqrt{\eta /\omega }$, $m=1$ and $1<\lambda <2$. The regular radial solution $b=qM((1+\lambda )/2,2,-q^2/2)$ has a $q^{-\lambda }$ tail and finite $I_b$. It gives $E\propto e^{2(2-\lambda )\omega t}$ and zero signed flux through every centered disk. Its absolute flux integral over the whole plane diverges, so the zero-flux statement here means the limit of these disk integrals. This example is not an isolated bounded-body dynamo: the velocity is unbounded at infinity and the field has an algebraic tail. The dynamo property concerns conversion of fluid energy into magnetic energy, not production of a nonzero signed magnetic flux; growth and its boundary assumptions must be specified.

For$x_0$ is an ordinary point when $P$ and $Q$ are analytic there, and a singular point otherwise. A singular point is regular singular when $(x-x_0)P(x)$ and $(x-x_0{)}^{2}Q(x)$ are analytic there. These are the ordinary-point and regular-singular criteria.

For Kummer's equation, substituteEquating the coefficient of $x^m$ givesTaking $c_0=1$,where $(a{)}_m=a(a+1)\cdots (a+m-1)$ is the rising factorial. Thus

Putting $y=x^{1-b}u$ and simplifying givesThereforeFor nonintegral $b$, the powers $x^0$ and $x^{1-b}$ at zero are distinct, so these solutions are linearly independent.

When $b\to 1$, both solutions tend to $M(x,a,1)$. Differentiate their difference with respect to $b$. Writing $M_a,M_b$ for derivatives with respect to the second and third arguments,Hence at $b=1$ one may takeas two linearly independent solutions. Their independence follows from the logarithmic term in the second solution.