# (e^ix)-1=(2ie^(ix/2))x(sin (x/2))

• MHB
• minachi12
If $r= 1$, the sum is $n$. Since $e^{ix}$ is never equal to $1$, the sum of this series is $\frac{1- e^{inx}}{1- e^{ix}}$. Now, $e^{inx}= (e^{ix})^n$. So, $1- e^{inx}= 1- (e^{ix})^n= (1- e^{ix})(1+ e^{ix}+...+ (e^{ix})^{n-1})$. Therefore $1+ e^{ix}+...+ (e^{ix})^{n-1}= \frac{1- minachi12 Show that: (e^ix)-1=(2ie^(ix/2))x(sin (x/2)) Calculate this sum Zn=1+e^(ix)+e^(2ix)+...+e^((n-1)ix) And deduce those values : Xn=1+cos(x)+cos(2x)+...+cos((n-1)x) Yn=sin(x)+sin(2x)+...+sin((n-1)x) minachi12 said: Show that: (e^ix)-1=(2ie^(ix/2))x(sin (x/2)) Is that "x" in the middle a multiplication sign? Don't do that! You need to know that$sin(x)= \frac{e^{ix}- e^{-ix}}{2i}$. So$sin(x/2)= \frac{e^{ix/2}- e^{-ix/2}}{2i}$,$sin(x/2)= \frac{e^{ix/2}- e^{-ix/2}}{2i}$,$2i e^{ix/2} sin(x/2)= e^{ix/2}(e^{ix/2}- e^{-ix/2})= e^{ix/2+ ix/2}- e^{ix/2- ix/2}= e^{ix}- 1$. Calculate this sum Zn=1+e^(ix)+e^(2ix)+...+e^((n-1)ix) And deduce those values : Xn=1+cos(x)+cos(2x)+...+cos((n-1)x) Yn=sin(x)+sin(2x)+...+sin((n-1)x) That's a geometric series,$\sum_{i= 0}^n r^i$with$r= e^{ix}$. The sum of such a geometric series is$\frac{1- r^n}{1- r}\$

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## 1. What is the meaning of the equation (e^ix)-1=(2ie^(ix/2))x(sin (x/2))?

This equation is a mathematical representation of Euler's formula, which states that e^(ix) can be expressed as cos(x) + isin(x), where i is the imaginary unit. The left side of the equation represents the complex number e^(ix) with a subtraction of 1, while the right side represents the product of 2i and e^(ix/2) multiplied by the sine of x/2.

## 2. What is the significance of the number e in this equation?

The number e, also known as Euler's number, is a mathematical constant that is approximately equal to 2.718. It is a fundamental constant in calculus and is often used in mathematical equations involving exponential growth and decay. In this equation, e represents the base of the natural logarithm and is raised to the power of the imaginary unit i multiplied by x.

## 3. How is this equation related to trigonometry?

This equation is related to trigonometry through Euler's formula, which relates complex numbers to trigonometric functions. In this equation, the sine and cosine functions are represented by the imaginary unit and the real number e, respectively. The equation also involves the use of half-angle identities, which are commonly used in trigonometric calculations.

## 4. Can this equation be simplified or solved for a specific value?

Yes, this equation can be simplified by expanding the left and right sides and using trigonometric identities. It can also be solved for a specific value by using algebraic methods or by plugging in values for x. However, the solution will still involve complex numbers and may not result in a simple numerical value.

## 5. What are some real-world applications of this equation?

This equation has many applications in physics and engineering, particularly in the fields of signal processing and electrical circuits. It is also used in quantum mechanics to describe the behavior of particles and waves. Additionally, this equation is used in the study of differential equations and complex analysis.

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