Greek Alphabet and Special Constants Formula Bank

Greek Alphabet:

The Greek alphabet consists of 24 letters and is commonly used in mathematics, physics, engineering, and other fields. Each letter has a lowercase and uppercase version. Here are the most frequently used Greek letters in scientific and mathematical contexts:

Special Constants

1. Pi (\( \pi \)):
\[
\pi = 3.14159\ 26535\ 89793\ \dots
\]

2. Euler’s number (\( e \)):
\[
e = 2.71828\ 18284\ 59045\ \dots = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n
\]

3. Euler’s constant (\( \gamma \)):
\[
\gamma = 0.57721\ 56649\ 01532\ 86060\ 6512 \dots
\]
\[
\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{k} – \ln n \right)
\]

4. \( e^\gamma \):
\[
e^\gamma = 1.78107\ 24179\ 90197\ 9852\ \dots
\]

5. \( \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \):
\[
\Gamma\left(\frac{1}{2}\right) = 1.77245\ 38509\ 05516\ 02729\ 8167 \dots
\]

6. \( \Gamma\left(\frac{1}{3}\right) \):
\[
\Gamma\left(\frac{1}{3}\right) = 2.67893\ 85347\ 07748\ \dots
\]

7. \( \Gamma\left(\frac{1}{4}\right) \):
\[
\Gamma\left(\frac{1}{4}\right) = 3.62560\ 99082\ 21908\ \dots
\]

8. Conversion between degrees and radians:
\[
1 \text{ radian} = \frac{180^\circ}{\pi} = 57.29577^\circ
\]
\[
1^\circ = \frac{\pi}{180} \text{ radians} = 0.01745 \text{ radians}
\]

In mathematics and physics, certain constants play fundamental roles. Here are a few notable ones:

1. Pi (π):
Approximate value: 3.14159…
Represents the ratio of a circle’s circumference to its diameter.

2. Euler’s Number (e):
 Approximate value: 2.71828…
The base of the natural logarithm, important in calculus and exponential growth/decay.

3. The Golden Ratio (φ):
Approximate value: 1.61803…
Appears in geometry, art, and architecture. The ratio between two quantities where the ratio of the larger to the smaller is the same as the whole to the larger.

4. The Speed of Light (c):
Exact value: 299,792,458 meters per second (m/s)
Fundamental constant in physics, representing the speed at which light travels in a vacuum.

5. Planck’s Constant (h):
Approximate value: \( 6.626 \times 10^{-34} \) Joule seconds (Js)
Relates the energy of a photon to its frequency, crucial in quantum mechanics.

6. Gravitational Constant (G):
Approximate value: \( 6.674 \times 10^{-11} \) \( \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)
Defines the strength of the gravitational force between two objects.

7. Avogadro’s Number (Nₐ):
Exact value: \( 6.022 \times 10^{23} \)
The number of atoms or molecules in one mole of a substance.