Editing Surreal number (section)

==Exponential function==
Based on unpublished work by [[Martin David Kruskal|Kruskal]], a construction (by [[transfinite induction]]) that extends the real [[exponential function]] {{math|exp(''x'')}} (with base {{mvar|[[E (mathematical constant)|e]]}}) to the surreals was carried through by Gonshor.<ref name=G1986 />{{rp|at=ch. 10}}

===Other exponentials===
The [[#Powers of ω|powers of {{mvar|ω}}]] function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-{{mvar|e}} exponential, and it is this function that is meant whenever the notation {{mvar|ω{{sup|x}}}} is used in the following.

When {{mvar|y}} is a dyadic fraction, the [[power function]] <math display=inline>x \in \mathbb{No}</math>, {{math|''x'' ↦ ''x''{{sup|''y''}}}} may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation {{math|1=''x''{{sup|''y''+''z''}} = ''x{{sup|y}}'' · ''x{{sup|z}}''}}, and where defined it necessarily agrees with any other [[exponentiation]] that can exist.

===Basic induction===
The induction steps for the surreal exponential are based on the series expansion for the real exponential,
<math display=block>\exp x = \sum_{n\ge 0} \frac{x^n}{n!}</math>
more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For {{mvar|x}} positive these are denoted {{math|[''x'']{{sub|''n''}}}} and include all [[partial sum]]s; for {{mvar|x}} negative but finite, {{math|[''x'']{{sub|2''n''+1}}}} denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For {{mvar|x}} negative infinite the odd-numbered partial sums are strictly decreasing and the {{math|[''x'']{{sub|2''n''+1}}}} notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.

The relations that hold for real {{math|{{nowrap|1=''x'' < ''y''}}}} are then{{ubl|{{nb5}}{{math|{{nowrap|1=exp ''x'' · [''y'' – ''x'']{{sub|''n''}} < exp ''y''}}}}}}and{{ubl|{{nb5}}{{nowrap|1={{math|exp ''y'' · [''x'' – ''y'']{{sub|2''n'' + 1}} < exp ''x''}},}}}}and this can be extended to the surreals with the definition

<math display=block>\exp z = \{0, \exp z_L \cdot [z-z_L]_n, \exp z_R\cdot[z-z_R]_{2n+1} \mid \exp z_R/[z_R-z]_n, \exp z_L/[z_L-z]_{2n+1} \}.</math>

This is well-defined for all surreal arguments (the value exists and does not depend on the choice of {{mvar|z{{sub|L}}}} and {{mvar|z{{sub|R}}}}).

===Results===
Using this definition, the following hold:{{efn|Even the most trivial-looking of these equalities may involve transfinite induction and constitute a separate theorem.}}
* {{math|exp}} is a strictly increasing positive function, {{math|1=''x'' < ''y'' ⇒ 0 < exp ''x'' < exp ''y''}}
* {{math|exp}} satisfies {{math|1=exp(''x'' + ''y'') = exp ''x'' · exp ''y''}}
* {{math|exp}} is a [[surjection]] (onto <math display=inline>\mathbb{No}_+</math>) and has a well-defined inverse, {{math|1=log = exp{{sup|–1}}}}
* {{math|exp}} coincides with the usual exponential function on the reals (and thus {{math|1=exp 0 = 1, exp 1 = ''e''}})
* For {{mvar|x}} infinitesimal, the value of the formal power series ([[Taylor expansion]]) of {{math|exp}} is well defined and coincides with the inductive definition
** When {{mvar|x}} is given in Conway normal form, the set of exponents in the result is well-ordered and the coefficients are finite sums, directly giving the normal form of the result (which has a leading {{math|1}})
** Similarly, for {{mvar|x}} infinitesimally close to {{math|1}}, {{math|log ''x''}} is given by power series expansion of {{math|''x'' – 1}}
* For positive infinite {{mvar|x}}, {{math|exp ''x''}} is infinite as well
** If {{mvar|x}} has the form {{mvar|ω{{sup|α}}}} ({{math|''α'' > 0}}), {{math|exp ''x''}} has the form {{mvar|ω{{sup|ω{{sup|β}}}}}} where {{mvar|β}} is a strictly increasing function of {{mvar|α}}. In fact there is an inductively defined bijection <math display=inline>g: \mathbb{No}_+ \to \mathbb{No} : \alpha \mapsto \beta</math> whose inverse can also be defined inductively
** If {{mvar|x}} is "pure infinite" with normal form {{math|1=''x'' = Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''a''{{sub|''α''}}}}}} where all {{math|1=''a''{{sub|''α''}} > 0}}, then {{math|1=exp ''x'' = ''ω''{{sup|Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''g''(''a''{{sub|''α''}})}}}}}}
** Similarly, for {{math|1=''x'' = ''ω''{{sup|Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''b''{{sub|''α''}}}}}}}}, the inverse is given by {{math|1=log ''x'' = Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''g''{{sup|–1}}(''b''{{sub|''α''}})}}}}
* Any surreal number can be written as the sum of a pure infinite, a real and an infinitesimal part, and the exponential is the product of the partial results given above
** The normal form can be written out by multiplying the infinite part (a single power of {{mvar|ω}}) and the real exponential into the power series resulting from the infinitesimal
** Conversely, dividing out the leading term of the normal form will bring any surreal number into the form {{math|1=(''ω''{{sup|Σ{{sub|''γ''<''δ''}}''t''{{sub|''γ''}}''ω''{{sup|''b''{{sub|''γ''}}}}}})·''r''·(1 + Σ{{sub|''α''<''β''}}''s''{{sub|''α''}}''ω''{{sup|''a''{{sub|''α''}}}})}}, for {{math|1=''a''{{sub|''α''}} < 0}}, where each factor has a form for which a way of calculating the logarithm has been given above; the sum is then the general logarithm
*** While there is no general inductive definition of {{math|log}} (unlike for {{math|exp}}), the partial results are given in terms of such definitions. In this way, the logarithm can be calculated explicitly, without reference to the fact that it's the inverse of the exponential.
* The exponential function is much greater than any finite power
** For any positive infinite {{mvar|x}} and any finite {{mvar|n}}, {{math|1=exp(''x'')/''x''{{sup|''n''}}}} is infinite
** For any integer {{mvar|n}} and surreal {{math|''x'' > ''n''{{sup|2}}}}, {{math|exp(''x'') > ''x''{{sup|''n''}}}}. This stronger constraint is one of the Ressayre axioms for the real [[exponential field]]<ref name=vdDE2001 />
* {{math|exp}} satisfies all the Ressayre axioms for the real exponential field<ref name=vdDE2001 />
** The surreals with exponential is an [[elementary extension]] of the real exponential field
** For {{math|''ε''{{sub|''β''}}}} an ordinal epsilon number, the set of surreal numbers with birthday less than {{math|''ε''{{sub|''β''}}}} constitute a field that is closed under exponentials, and is likewise an elementary extension of the real exponential field

===Examples===
The surreal exponential is essentially given by its behaviour on positive powers of {{mvar|ω}}, i.e., the function {{tmath|g(a)}}, combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition, {{tmath|1=g(a) = a}} holds for a large part of its range, for instance for any finite number with positive real part and any infinite number that is less than some iterated power of {{mvar|ω}} ({{mvar|ω{{sup|ω{{sup|·{{sup|·{{sup|ω}}}}}}}}}} for some number of levels).
* {{math|1=exp ''ω'' = ''ω''{{sup|''ω''}}}}
* {{math|1=exp ''ω''{{sup|1/''ω''}} = ''ω''}} and {{math|1=log ''ω'' = ''ω''{{sup|1/''ω''}}}}
* {{math|1=exp (''ω'' · log ''ω'') = exp (''ω'' · ''ω''{{sup|1/''ω''}}) = ''ω''{{sup|''ω''{{sup|1 + 1/''ω''}}}}}}
** This shows that the "power of {{mvar|ω}}" function is not compatible with {{math|exp}}, since compatibility would demand a value of {{mvar|ω{{sup|ω}}}} here
* {{math|1=exp ''ε''{{sub|0}} = ''ω''{{sup|''ω''{{sup|''ε''{{sub|0}} + 1}}}}}}
* {{math|1=log ''ε''{{sub|0}} = ''ε''{{sub|0}} / ''ω''}}

===Exponentiation===
A general exponentiation can be defined as {{nowrap|{{math|1=''x{{sup|y}}'' = exp(''y'' · log ''x'')}}}}, giving an interpretation to expressions like {{nowrap|{{math|1=2{{sup|''ω''}} = exp(''ω'' · log 2) {{if mobile|<br />|}}= ''ω''{{sup|log 2 · ''ω''}}}}}}. Again it is essential to distinguish this definition from the "powers of {{mvar|ω}}" function, especially if {{mvar|ω}} may occur as the base.