Common gain ,Relation between alpha α beta β gamma γ

Common gain

Current gain or current amplification factor is defined by the ratio of output current to input current when output voltage is kept constant

CB Configuration
\( \left. α=\frac{ΔI_{C}}{ΔI_{E}}  \right |_{V_{CB}=constant } \)
CE Configuration
\( \left. β=\frac{ΔI_{C}}{ΔI_{B}}  \right |_{V_{CE}=constant } \)
CC Configuration 

\( \left. γ=\frac{ΔI_{E}}{ΔI_{B}}  \right |_{V_{CE}=constant } \)

Relation between alpha α beta β gamma γ

We know that 

\(I_E=I_B+I_C\)→1

dividing it by  \(I_C\) gives

\(\frac{I_E}{I_C}=\frac{I_B}{I_C}+\frac{I_C}{I_C}\)
\(\frac{1}{α }=\frac{1}{β }+\frac{1}{1}=\frac{1}{β }+1\)
\(α =\frac{β}{β+1 }\)  

\(\frac{1}{β }=\frac{1}{α }-1\) 
\(\frac{1}{β }=\frac{1-α}{α }\) 
\(β =\frac{α }{1-α}\) 

dividing it by  \(I_B\) gives

 \(\frac{I_E}{I_B}=\frac{I_B}{I_B}+\frac{I_C}{I_B}\)
\(γ=1+β\)
\(β=γ-1\)