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This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative . Also see the Mathematical Association of America Math DL review , and the Amazon reviews .
Consider the function on the right hand side (RHS) f(x) = cos( x ) + i sin( x ) Differentiate this function f ' (x) = -sin( x ) + i cos( x) = i f(x) So, this function has the property that its derivative is i times the original function. What other type of function has this property? A function g(x) will have this property if dg / dx = i g This is a differential equation that can be solved with seperation of variables (1/g) dg = i dx (1/g) dg = i dx ln| g | = i x + C | g | = e i x + C = e C e i x | g | = C 2 e i x g = C 3 e i x So we need to determine what value (if any) of the constant C 3 makes g(x) = f(x). If we set x=0 and evaluate f(x) and g(x), we get f(x) = cos( 0 ) + i sin( 0 ) = 1 g(x) = C 3 e i 0 = C 3 These functions are equal when C 3 = 1. Therefore, cos( x ) + i sin( x ) = e i x (This is the usual justification given in textbooks.)
In information theory , entropy is a measure of the uncertainty in a random variable . [ 1 ] In this context, the term usually refers to the Shannon entropy , which quantifies the expected value of the information contained in a message. [ 2 ] Entropy is typically measured in bits , nats , or bans . [ 3 ] Shannon entropy is the average unpredictability in a random variable, which is equivalent to its information content . The concept was introduced by Claude E. Shannon in his 1948 paper " A Mathematical Theory of Communication ". [ 4 ] Shannon entropy provides an absolute limit on the best possible lossless encoding or compression of any communication, assuming that [ 5 ] the communication may be represented as a sequence of independent and identically distributed random variables .