Exponential of Sum/Complex Numbers/Corollary

Corollary to Exponential of Sum: Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\exp z$ be the exponential of $z$.

Then:

$\map \exp {z_1 - z_2} = \dfrac {\exp z_1} {\exp z_2}$

Proof

By Exponential of Sum: Complex Numbers:

$\map \exp {z_1 - z_2} = \exp z_1 \, \map \exp {-z_2}$

By Reciprocal of Complex Exponential:

$\dfrac 1 {\exp z_2} = \map \exp {-z_2}$

Combining these two, we obtain the result:

$\map \exp {z_1 - z_2} = \dfrac {\exp z_1} {\exp z_2}$

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $3$