Elementary Functions of Complex Numbers

Trigonometry and Hyperbolic Functions in 

Complex Numbers

Trigonometry Functions:

`e^(ix) = (\cos x + i\sin x)`

and    `e^(-ix) = (\cos x - i\sin x)`           (by Euler’s Formula)

Adding & subtracting above, we have

`\cos x =\frac{e^(ix) + e^(-ix)}{2}`       and      `\sin x =\frac{e^(ix) - e^(-ix)}{2i}` 

`\implies`    `\cos z =\frac{e^(iz) + e^(-iz)}{2}`     ,      `\sin z =\frac{e^(iz) - e^(-iz)}{2i}` 

Furthermore,

`\tan z = \frac{\sin z}{\cosz}       ,        \cot z = \frac{\cos z}{\sinz}`

`\sec z = \frac{1}{\cosz}       ,        \csec z = \frac{1}{\sinz}`

Note: Since `e^z` is entire function, therefore 𝑠𝑖𝑛𝑧 & π‘π‘œπ‘ π‘§ both are entire functions.

Hyperbolic Functions 

For any complex number 𝑧 = π‘₯ + 𝑦i

`\sinh z =\frac{e^z - e^(-z)}{2}`     and      `\cosh z =\frac{e^z + e^(-z)}{2}`

Relation between Complex Trigonometric and Hyperbolic Functions

`\sin iz = i \sin hz              ,           \cos iz = \coshz`
`\sinhiz = i \sinz                ,           \coshiz = \cosz`



Examples:



Example 7: Express cosh (3 + 4𝑖) in the form 𝑒 + 𝑖𝑣. 

Solution: 
π‘π‘œπ‘  β„Ž(3 + 4𝑖) = π‘π‘œπ‘  𝑖(3 + 4𝑖) 
 = π‘π‘œπ‘ (−4 + 3𝑖) 
 = π‘π‘œπ‘ (−4) π‘π‘œπ‘ (3𝑖) − 𝑠𝑖𝑛(−4) 𝑠𝑖𝑛 (3𝑖) 
= (−0.6536)(10.067) − (0.7568). π‘–π‘ π‘–π‘›β„Ž(3) 
 = (−0.6536)(10.067) − (0.7568). (10.0179)𝑖 π‘π‘œπ‘  β„Ž(3 + 4𝑖) 
= −6.5797 − 7.5815𝑖