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E–Z configuration, or the E–Z convention, is the IUPAC preferred method of describing the absolute stereochemistry of double bonds in organic chemistry.It is an extension of cis–trans isomer notation (which only describes relative stereochemistry) that can be used to describe double bonds having two, three or four substituents.
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
Very often, cis–trans stereoisomers contain double bonds or ring structures. In both cases the rotation of bonds is restricted or prevented. [4] When the substituent groups are oriented in the same direction, the diastereomer is referred to as cis, whereas when the substituents are oriented in opposing directions, the diastereomer is referred to as trans.
Since the multiplicative property still works for the complex exponential function, e z = e z+2kiπ, for all complex z and integers k. So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2iπ at will.
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
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One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations.The solution of = (), =, where A is a constant matrix and y is a column vector, is given by =.