Exponential: Difference between revisions

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:''This page describes the monadic arithmetic function. For the dyadic function, see [[Power (function)]].''
:''This page describes the monadic arithmetic function. For the dyadic function, see [[Power (function)]].''


{{Built-in|Exponential|*}} is a [[monadic]] [[scalar function]] which computes the [[wikipedia:exponential function|exponential function]] (i.e. the power of [[wikipedia:e (mathematical constant)|Euler's constant e]]) of the [[argument]]. Exponential shares the [[glyph]] <source lang=apl inline>*</source> with the dyadic arithmetic function [[Power]].
{{Built-in|Exponential|*}} is a [[monadic]] [[scalar function]] which computes the [[wikipedia:exponential function|exponential function]] (i.e. the power of [[wikipedia:e (mathematical constant)|Euler's constant e]]) of the [[argument]]. Exponential shares the [[glyph]] <source lang=apl inline>*</syntaxhighlight> with the dyadic arithmetic function [[Power]].


== Examples ==
== Examples ==
Line 10: Line 10:
       *1
       *1
2.718281828
2.718281828
</source>
</syntaxhighlight>


On APL implementations that support [[complex]] numbers, one can demonstrate [[wikipedia:Euler's identity|Euler's identity]] (with the help of [[Pi Times]] <source lang=apl inline>○</source>):
On APL implementations that support [[complex]] numbers, one can demonstrate [[wikipedia:Euler's identity|Euler's identity]] (with the help of [[Pi Times]] <source lang=apl inline>○</syntaxhighlight>):


<source lang=apl>
<source lang=apl>
       1+*○0J1
       1+*○0J1
0
0
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


== Properties ==
== Properties ==


Exponential is a special case of [[Power]] with the default left argument of e (<source lang=apl inline>*1</source>).
Exponential is a special case of [[Power]] with the default left argument of e (<source lang=apl inline>*1</syntaxhighlight>).


<source lang=apl>
<source lang=apl>
       ((*1)∘* ≡ *) 0 1 ¯1 0J1
       ((*1)∘* ≡ *) 0 1 ¯1 0J1
1
1
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


Exponential and [[Natural Logarithm|natural log]] <source lang=apl inline>⍟</source> are inverses of each other, except where the natural log is undefined.
Exponential and [[Natural Logarithm|natural log]] <source lang=apl inline>⍟</syntaxhighlight> are inverses of each other, except where the natural log is undefined.


<source lang=apl>
<source lang=apl>
Line 35: Line 35:
       (⊢ ≡ *∘⍟) 1 ¯1 0J1 ⍝ natural log of zero is undefined
       (⊢ ≡ *∘⍟) 1 ¯1 0J1 ⍝ natural log of zero is undefined
1
1
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


== See also ==
== See also ==

Revision as of 21:46, 10 September 2022

This page describes the monadic arithmetic function. For the dyadic function, see Power (function).
*

Exponential (*) is a monadic scalar function which computes the exponential function (i.e. the power of Euler's constant e) of the argument. Exponential shares the glyph <source lang=apl inline>*</syntaxhighlight> with the dyadic arithmetic function Power.

Examples

Euler's constant itself can be obtained by supplying 1 as the argument.

<source lang=apl> 

     *1

2.718281828 </syntaxhighlight>

On APL implementations that support complex numbers, one can demonstrate Euler's identity (with the help of Pi Times <source lang=apl inline>○</syntaxhighlight>):

<source lang=apl> 

     1+*○0J1

0

</syntaxhighlight>

Works in: Dyalog APL

Properties

Exponential is a special case of Power with the default left argument of e (<source lang=apl inline>*1</syntaxhighlight>).

<source lang=apl> 

     ((*1)∘* ≡ *) 0 1 ¯1 0J1

1

</syntaxhighlight>

Works in: Dyalog APL

Exponential and natural log <source lang=apl inline>⍟</syntaxhighlight> are inverses of each other, except where the natural log is undefined.

<source lang=apl> 

     (⊢ ≡ ⍟∘*) 0 1 ¯1 0J1

1 

     (⊢ ≡ *∘⍟) 1 ¯1 0J1 ⍝ natural log of zero is undefined

1

</syntaxhighlight>

Works in: Dyalog APL

See also

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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