Basic Examples (3) 

A range of positive transcendental numbers:

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Negative and positive transcendental numbers:

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A transcendental range with non-unit step:

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Scope (4) 

A symmetric transcendental range with step:

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A transcendental range with negative step:

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Require a lower bound on the steps:

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A very large transcendental range generated efficiently:

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2.338204`

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Options (17) 

Method (8) 

A range in terms of logarithms:

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Ranges in terms of hyperbolic transcendental functions:

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Ranges in terms of inverse hyperbolic functions:

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Ranges in terms of trigonometric functions:

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Ranges in terms of inverse trigonometric functions:

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Ranges in terms of Power (this requires the setting "GeneratorsDomain"→Algebraics to get non-trivial results):

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A range in terms of a list of different types:

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A range in terms of all previous types combined:

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GeneratorsDomain (2) 

A transcendental range generated by non rationals Algebraics:

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Compare with the corresponding default range in terms of the default Rationals:

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FareyRange (2) 

Generate steps according to the FareySequence:

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Compare with the default transcendental range of step 1/2:

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Also the default transcendental range of step 1/3:

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It turns out that the transcendental Farey range does not correspond to simply combining ordinary ranges with more steps, but to a superset of it:

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Create a transcendental Farey range including root arguments:

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FormulaComplexity (3) 

Reduce the output range by setting a threshold on the complexity:

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Compare with the default range with no complexity threshold:

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Compare with the same range restricted by the fourth argument (lower bound on steps):

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WorkingPrecision (2) 

In certain cases the generated symbolic expressions may be found to collide in terms of their N numerical values:

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To distinguish them, WorkingPrecision may be optionally set to a higher value:

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Applications (4) 

Search and find closed forms in terms of exponentials:

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Find closed forms in terms of all elementary transcendental functions:

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Find closed forms in terms of common mathematical functions composed to exponentials:

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Search and find closed forms in terms of common functions composed to any of the available transcendental functions:

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Properties and Relations (2) 

TranscendentalRange[x,y,z] can be simply generated through the Outer product of the function #1 Exp[#2]&, with both arguments running over Range[x,y,z], suitably restricted to the elements within the bounds x, y, not belonging to the Algebraics and sorted, as follows:

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The actual implementation of TranscendentalRange[x,y,z] is much more efficient than Outer over large exponentially increasing ranges:

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Possible Issues (2) 

By definition, the arguments must be all algebraic numbers:

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Bypassed this limitation by increasing the arguments:

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Neat Examples (2) 

Generate a mixed large transcendental range with all the methods available:

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Pick a random element of it and verify experimentally the property of being transcendental, that is that its RootApproximant expressions do not converge at any given precision:

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It is interesting to do the same computation on algebraic numbers. Install this paclet to generate more general algebraic numbers:

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Generate a large algebraic range and pick a random algebraic number in it:

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We find for this a quite rapid convergence beyond 64 approximation digits:

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Take also the (nested) algebraic of this range which is

Find the most complex in terms of LeafCount:

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The convergence of RootApproximant is slower but necessarily happens after about 512 approximation digits:

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Note this agrees with the minimal polynomial:

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Notice the function Element can confirm the algebraic character but not always the transcendental one:

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It should be emphasized though that while these are easy computational investigations within the Wolfram Language, a proper rigorous mathematical proof is necessary to establish the transcendence character of a number (that is equivalent to evaluating it at infinite precision).