Maximum curves and isolated points of entire functions

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

Study of Anisotropic Conductive Adhesive Joint Behaviour under 3-Point Bending

Flip chip interconnections using anisotropic conductive film (ACF) are now a very attractive technique for electronic packaging assembly. Although ACF is environmentally friendly, many factors may influence the reliability of the final ACF joint. External mechanical loading is one of these factors. Finite element analysis (FEA) was carried out to understand the effect of mechanical loading on the ACF joint. A 3-dimensional model of adhesively bonded flip chip assembly was built and simulations were performed for the 3-point bending test. The results show that the stress at its highest value at the corners, where the chip and ACF were connected together. The ACF thickness was increased at these corner regions. It was found that higher mechanical loading results in higher stress that causes a greater gap between the chip and the substrate at the corner position. Experimental work was also carried out to study the electrical reliability of the ACF joint with the applied bending load. As per the prediction from FEA, it was found that at first the corner joint failed. Successive open joints from the corner towards the middle were also noticed with the increase of the applied load.