Arguments of Transcendental Functions
For a physically meaningful expression, the complete argument of exp, log, sin, cos, or any power-series-defined transcendental function must be dimensionless because terms in its series expansion must be addable.
Why this shows up in the exam
Checking wave phases · Validating decay laws · Interpreting statistical factors
Learn the idea
The input to sine, cosine, exponential, and logarithm functions must be dimensionless. A universal function cannot depend on whether length was entered in metres or centimetres, so its complete argument must be a pure number.
🧠 Memory hook: Functions eat pure numbers, not metres or seconds.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- [omega t + phi] = 1 — phase in a trigonometric function is dimensionless
- [x/lambda] = 1 — a spatial exponential or sinusoid needs a pure ratio
- [E/(k_B T)] = 1 — Boltzmann exponential argument is dimensionless
How to approach it
- 1Circle every transcendental function
- 2Set its full argument dimensionless
- 3Solve for any unknown constant dimensions
Common slip-ups that cost marks
- •Checking only one term of a summed phase
- •Assigning a dimension to pi
- •Ignoring a hidden scale inside a logarithm
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A quantity Q is defined as Q = force^1 / speed^2. Which dimensional formula represents Q?
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