# Laplace Transformation

[latexpage] HISTORY Integral transformation data back to the Work of Leonard Euler (1763 & 1769), who considered them essentially in the form of the inverse Laplace transform in solving second-order, linear ordinary differential equations. Even Laplace, in his great work, “Theorie Analytique...

# Compact Sets and Continuous Functions on Compact Sets

[latexpage] INTRODUTION In advance analysis, the notion of 'Compact set' is of paramount importance. In $\mathbb{R}$, Heine-Borel theorem provides a very simple characterization of compact sets. The definition and techniques used in connection with compactness of sets in $\mathbb{R}$ are...

# Fourier Series

INTRODUCTION The study of a special type of series whose terms are trigonometric functions of a variable was started in the 18th century when J.B.J. Fourier (1768-1830) was successful in his attempt to prove that an arbitrary function given in the interval $[-\pi,\pi]$ can be expressed under...

# All The Logarithm Rules You Know and Don’t Know About

[latexpage] Definition The logarithm function is one of the most known and used functions in mathematics and other fields like physics, finance, chemistry, ...etc. this article is logarithmic heaven where you can learn many things about logarithmic functions and where you can find a large list of...

# Riemann Integral

[latexpage] Introduction: The story of integration started with I. Newton, when in the late 1660 he invented the method of inverse tangents to find areas under curves. In 1680, G. Leibnitz discovered the process of finding tangent line to find area. Thus, they had discovered the integration, being...

# Lebesgue Outer Measure

[latexpage] Introduction: The Riemann integral of a bounded function over a closed, bounded interval is defined using approximations of the function that are associated with partitions of its domain into finite collections of subintervals. The generalization of the Riemann integral to the Lebesgue...