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Register a new userOn a family of curious integrals suggested by Stellar Dynamics
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While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)equivint_0^1{{rm K}(k) kover (z+k^2)^{n+3/2}}dk = {(-2)^nover (2n+1)!!} {d^nover dz^n} {{rm ArcCot}sqrt{z}oversqrt{z(z+1)}},quad z>0,$$ where $n=0,1,2,3...$, and ${rm K}(k)$ is the complete elliptic integral of first kind.
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While investigating the generalization of the Chandrasekhar (1943) dynamical friction to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error function occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010). Here we show that both the integrals are very special cases of the family of (real) functions $$ I(lambda,mu, u; z) :=int_0^zx^{lambda},Enu(x^{mu}),dx= {gammaleft({1+lambdaovermu},z^{mu}right) + z^{1+lambda}Enu(z^{mu})over 1+lambda + mu ( u -1)}, quad mu>0,quad zgeq 0, eqno (1) $$ where $Enu$ is the Exponential Integral, $gamma$ is the incomplete Euler gamma function, and for existence $lambda >max left{-1,-1- mu( u -1)right}$. Only in one of the consulted tables a related integral appears, that with some work can be reduced to eq.~(1), while computer algebra systems seem to be able to evaluate the integral in closed (and more complicated) form only provided numerical values for some of the parameters are assigned. Here we show how eq.~(1) can in fact be established by elementary methods.
The notion of integral Bailey pairs is introduced. Using the single variable elliptic beta integral, we construct an infinite binary tree of identities for elliptic hypergeometric integrals. Two particular sequences of identities are explicitly described.
We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp condition for the kernels. Also, we prove some weighted $L^p$ inequalities for the operators.
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9s is also a convergent series. We show how to compute sums of Irwins series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.007x10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol.
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
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