# Equating coefficients of complex exponentials

• I
• AtoZ
In summary, the conversation discusses an equation with constants and an ansatz that must be equated to coefficients. After equating the coefficients, it is determined that the denominator must be written in full to be accurate, and the fraction must be split into two parts. The incorrect statement that ##\frac{a+b}{c+d}## is equal to ##\frac a c + \frac b d## is also addressed.
AtoZ
I have an equation that looks like

##i\dot{\psi_n}=X~\psi_n+\frac{C~\psi_n+D~a~\psi^\ast_{n+1}+E~b~\psi_{n+1}}{1+\beta~(D~\psi^\ast_{n+1}+E~\psi_{n+1})}##

where ##E,b,D,a,C,X## are constants. I have the ansatz

##\psi_n=A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast}##, ##x## and ##A_n,B_n## are complex. I have to equate coefficients of ##e^{ixt}## and ##e^{-itx^\ast}##, I get

##-xA_n~e^{ixt}+x^*B^\ast_n~e^{-itx^*}=\left[X~(A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast})+\frac{C~(A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast})+D~a~(A_{n+1}^*~e^{-itx^*}+B_{n+1}~e^{ixt})+E~b~(A_{n+1}~e^{ixt}+B^*_{n+1}~e^{-itx^*})}{1+\beta~[D~(A_{n+1}^*~e^{-itx^*}+B_{n+1}~e^{ixt})+E~(A_{n+1}~e^{ixt}+B^*_{n+1}~e^{-itx^*})]}\right]##

Now to equate coefficients of say ##e^{ixt}##, I get

##-xA_n=XA_n+\frac{C~A_n+D~a~B_{n+1}+E~b~A_{n+1}}{1+\beta(D~B_{n+1}+E~A_{n+1})}## is true? or the denominator has to be written in full?

AtoZ said:
or the denominator has to be written in full?
Even worse, you'll have to split the fraction properly into one part proportional to ##e^{ixt}## and one proportional to ##e^{itx^*}##, if this is possible at all.

AtoZ
means the one I wrote is incorrect?

Yes.

##\frac{a+b}{c+d} \neq \frac a c + \frac b d##

AtoZ

## What is the purpose of equating coefficients of complex exponentials?

The purpose of equating coefficients of complex exponentials is to solve equations involving complex numbers and find the values of the unknown variables. It is a useful technique in many areas of science, such as electrical engineering, physics, and mathematics.

## How is the process of equating coefficients of complex exponentials carried out?

The process involves equating the real and imaginary parts of the complex exponential expression to the corresponding real and imaginary coefficients. This results in a system of equations that can be solved simultaneously to find the values of the unknown variables.

## What are the common challenges when equating coefficients of complex exponentials?

One common challenge is identifying the real and imaginary parts of the complex exponential expression. Another challenge is dealing with equations that have multiple variables and coefficients, which require a systematic approach to solve.

## Can equating coefficients of complex exponentials be applied to any type of equation?

No, equating coefficients of complex exponentials is specifically used for equations involving complex numbers and exponential terms. It is not applicable to equations with other types of functions, such as polynomials or trigonometric functions.

## What are some real-life applications of equating coefficients of complex exponentials?

Equating coefficients of complex exponentials is used in many scientific fields, including circuit analysis, quantum mechanics, and signal processing. It is also used in practical applications such as designing electronic circuits, analyzing electromagnetic fields, and solving differential equations in physics and engineering problems.

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