The Journal of the Indian Mathematical Society

M. Bhimasena Rao , M. Venkatarama Iyer

Abstract

Let     F ( z , n )= ∫ ^∞ ₀ e^-xz M_x (x) dx,

where the function M_x(x)* is such that the above integral is convergent and that as x ⃗ ∞, M_x (x) e ^-xz ⃗ 0, for all integral values of n ≥ 1 (a condition satisfied in the case of the several functions which were obtained as particular cases of the general M -function).

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M. Bhimasena Rao , M. Venkatarama Iyer

Abstract

In a paper on ' Determinants involving specified numbers,' three persymmetric determinants whose elements are integrals of Legendre's Polynomials were taken and their values given without proof.† In this paper, the missing proof and a general mode of evaluation are supplied along with a few more new results.

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C. Krishnamachari , M. Bhimasena Rao

Abstract

[We have contributed in a series of papers our researches relating to the evaluation of certain classes of determinants to the Journal of the Indian Mathematical Society. The main results relating to persymmetrics have been communicated to the London Mathematical Society. In the course of our work, we have discovered a very elegant theorem, which has many interesting applications. In particular, we find that most of the results proved in our previous papers can be deduced from this theorem. This paper contains this theorem and some of its applications].

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C. Krishnamachari , M. Bhimasena Rao

Abstract

Hancock in his Theory of Elliptic Functions, Vol. I, page 252, gives the first five co-efficients in the expansion of cn (x,k), as having been calculated by Gudermann in Crelle, Bd. XIX, p. 80. The authors pointed out in their paper : "Determinants involving Specified Numbers," Vol. XIV, No. 4, J. I. M. S., pp. 122-138, that the ascending Σ-table formed with a_2n-1 = (2n-1)², a_2n = (2n)²k² gives the co-efficients in the expansion of cn(x,k) and that with a_2n-1 = (2n-1)²k², a_2n = (2n)² gives the co-efficients in the expansion of dn (x,k)-Ibid, pp. 133, § 6.

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M. Bhimasena Rao

Abstract

The object of this paper is to determine the values of x and z for a given value of sin 2y, by the methods of Elementary Geometry. The rationalised trigonometric form stated above will be found to be best adapted for this purpose. The method of solution by the use of Elliptic Functions is briefly indicated at the end of the paper.

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M. Bhimasena Rao , M. Venkatarama Ayyar

Abstract

This paper deals with the evaluation of certain infinite series and products. A few of them have already been evaluated by Dr. Glaishei and others; but, their mode of evaluation is complex, involving a knowledge of elliptic functions and modular equations. The method adopted here is fairly elementary, consisting of simple transformations of two well-known infinite integrals and is believed to be new.

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M. Bhimasena Rao

Abstract

Twin points with respect to a triangle ABC, are diametrically opposite points on a rectangular hyperbola circumscribing the triangle. Since H, the orthocentre of the triangle, lies on every circumscribing rectangular hyperbola, points which are twins with respect to the triangle ABC are also twins with respect to the triangles HBC, HCA, HAB.

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M. Bhimasena Rao , M. Venkatarama Ayyar

Abstract

In Part I of this paper several general results were obtained wherein a was put equal to πr (cos θ + i sin θ) and real and imaginary parts were separated. The equated real parts led to the evaluation of several theta-products and a few series.

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M. Venkatarama Ayyar ¹, M. Bhimasena Rao ¹

Affiliations
1 Bangalore, IN

Abstract

Our object in this paper is to show the underlying unity in all the six examples of Ramanujan, reducing them and his algebraic identity to one type and to extend the work to several more types.

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M. Bhimasena Rao

Abstract

Let S be the centre of a conic touching the sides of a triangle ABO at D,E,F. We shall call the circle through D,E,F, the contact circle of the point S with respect to the triangle ABC.

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M. Bhimasena Rao

Abstract

Let ABC and A'B'C' be two triangles, and P and P' be the mean centres of A,B,C and A', B', C' respectively for !any common system of multiples. The points P and P' may be said to be corresponding points of the triangles and the point which is its own correspondent may be called the double point of the triangles.

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