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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.
Because the cis–trans and E–Z systems compare different groups on the alkene, it is not strictly true that Z corresponds to cis and E corresponds to trans. For example, trans -2-chlorobut-2-ene (the two methyl groups, C1 and C4, on the but-2-ene backbone are trans to each other) is ( Z )-2-chlorobut-2-ene (the chlorine and C4 are together ...
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 .
Therefore, the absolute value of e z can be 1 only if e x is 1; since x is real, that happens only if x = 0. Therefore z is purely imaginary and cos(y) + i sin(y) = 1. Since y is real, that happens only if cos(y) = 1 and sin(y) = 0, so that y is an integer multiple of 2 π. Consequently the singular points of this function occur at
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We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the conjugate transpose of X. If Y is invertible then e YXY −1 = Ye X Y −1. The next key result is this one: