Complex numbers: Complex functions
Complex sine and cosine
The trigonometric functions can be expressed in powers of #\e# with imaginary exponents:
Cosine and sine in terms of imaginary powers of e \[ \begin{array}{rcl}\cos(\varphi) &=& \frac{\e^{\varphi\cdot\ii}+\e^{-\varphi\cdot\ii}}{2}\\ \sin(\varphi) &=& \frac{\e^{\varphi\cdot\ii}-\e^{-\varphi\cdot\ii}}{2\ii}\end {array}\]
Proof: Double application of the Formula of Euler gives:
\[ \begin{array}{rcl}\frac{\e^{\varphi\cdot\ii}+\e^{-\varphi\cdot\ii}}{2}&=&\frac{\cos(\varphi )+\sin(\varphi )\cdot\ii+\cos(-\varphi )+\sin(-\varphi )\cdot\ii}{2}\\&=&\frac{\cos(\varphi )+\sin(\varphi )\cdot\ii+\cos(\varphi )-\sin(\varphi )\cdot\ii}{2}\\&=&\frac{2\cos(\varphi )}{2}\\&=&\cos(\varphi )\end {array}\] This proves the first equality. The proof of the second one works in the same manner.
We use these equalities to define to the sine and cosine for all complex numbers #z#.
Complex sine and cosine \[ \begin{array}{rcl}\cos(z) &=& \frac{1}{2}\Big(\e^{z\cdot\ii}+\e^{-z\cdot\ii}\Big)\\\\ \sin(z) &=& \frac{1}{2\ii}\Big(\e^{z\cdot\ii}-\e^{-z\cdot\ii}\Big)\end {array}\]
If #z# is real, then the formulas above show that the values are the ones of the known sine and cosine. Hence, we are dealing here with an extension of the definition of the domain #\mathbb R# of the real numbers to the domain #\mathbb C# of the complex numbers.
These functions have a few properties which are well-known from the real case. First, they are once again periodically. Just as in the real case a complex function #f(z)# is called periodic with period #a# if #f(z+a)=f(z)# for all #z#.
Periodicity of the complex exponential and trigonometric functions
- The complex exponential function #\exp# is periodic with period #2\pi\cdot\ii#.
- The complex sine and cosine, #\sin# and #\cos#, are periodic with period #2\pi#.
The periodicity of the sine follows from the following calculation: \[ \begin{array}{rcl}\sin(z+2\pi)&=&\frac{1}{2\ii}\left(\e^{(z+2\pi)\cdot\ii}-\e^{-(z+2\pi)\cdot\ii}\right)\\&&\phantom{xyz}\color{blue}{\text{definition}}\\&=&\frac{1}{2\ii}\left(\e^{z\cdot\ii}-\e^{-z\cdot\ii}\right)\\&&\phantom{xyz}\color{blue}{\text{periodicity of }\exp}\\&=&\sin(z) \end {array}\]
The proof that the cosine is periodic with period #2\pi# works in the same manner.
Also, the following well-known formula is valid for all complex numbers.
\[\sin^2 (z)+\cos^2(z)=1\ \ \ \forall z\in \mathbb{C}\tiny.\]
The equality #\sin^2 (z)+\cos^2 (z)=1# can be found by entering the definitions of #\sin (z)# and #\cos (z)#.
The addition formulas also remain valid. However, the absolute value of the complex cosine is not always less than or equal to 1, as for the real cosine.
We have to solve the following equation with unknown:
\[\frac12 \left(\e^{z\cdot\ii}+\e^{-z\cdot\ii}\right) = 9 \tiny .\]Write #w=\e^{z\cdot\ii}#. Then we have #w \neq 0# because the range of #\exp# is equal to #\mathbb{C}\setminus\{0\}#. Rewriting the equation in #w# gives the following equation and derivation in #w#.
\[\begin{array}{rcl} w+\frac{1}{w} &=& 18\\
w^2-18 w+1&=&0\\ &&\phantom{xyzuvw}\color{blue}{\text{multiplied by }w}\\
(w-9)^2&=&80\\ &&\phantom{xyzuvw}\color{blue}{\text{completing the square}}\\
w&=&9\pm {4\sqrt{5}}\\&&\phantom{xyzuvw}\color{blue}{\text{taking the real root}}\\
\end{array}
\]
Of which follows
\[\begin{array}{rcl}
\e^{-\Im(z)}&=&\e^{\Re(z\cdot\ii)} = \left| \e^{z\cdot\ii}\right|= |w|=9\pm{4\sqrt{5}}\ ,\ {\rm hence,}\\ \Im (z)&=&-\ln (9\pm {4\sqrt{5}})\\ \Re (z)&=&-\Im(z\cdot\ii) =-\arg \left(\e^{z\cdot\ii}\right)=-\arg( w)=0 \pmod{2\pi} ,\ {\rm dus}\\ \Re(z)&=&2k\cdot\pi\phantom{xx}\left( k\in\mathbb{Z}\right)\end{array}\]
The solutions to the solution are \[\begin{array}{rcl}z & =&2k\cdot \pi-\ln (9+4\sqrt{5})\cdot\ii\phantom{xx}\left( k\in\mathbb{Z}\right)\\ &\lor&\\z & =&2k\cdot \pi-\ln (9-4\sqrt{5})\cdot\ii\phantom{xx}\left(k\in\mathbb{Z} \right)\end{array}\]
The absolute value of the complex cosine is not always smaller than or equal to 1, as is the case for the real cosine.
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