# Frobenius Series: Why c0 ≠ 0?

• IniquiTrance
In summary, the first term c0 in a Frobenius series cannot equal 0 because it is necessary for a0 to not vanish in order for the formal sum to converge and solve the differential equation. If a0 were to vanish, the result would not be interesting as the coefficient A could be any number.

#### IniquiTrance

Why do we assume that the first term c0 in a frobenius series
cannot equal 0?

Thanks!

Latex seems to be misbehaving, so I'll write in plain text:

In the Frobenius substitution, the x dependence of the first term is already factored out:

y(x) = x^r Sum (a_k x^k)

So, the first term in the series is actually

a_0 x^r

and when we plug the series into the differential equation, the question we are asking is "What is the smallest r for which a_0 does not vanish?" The answer is given by the indicial equation.

After solving the indicial equation for r, we are then equipped to ask the next question: "Given that a_0 does not vanish, can I find some sequence a_k such that my formal sum converges and solves the differential equation?"

Thanks for the response. Why is it necessary though that a0 not vanish?

IniquiTrance said:
Thanks for the response. Why is it necessary though that a0 not vanish?

Let say the first term that we obtained on substituting the Frobenius series into the DE as

Aa0xr + ... is identically zero.

This implies Aa0=0.
We may assume a0 to be zero or nonzero. But if it is zero then A can be any number. Not an interesting result.

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## 1. Why is c0 not equal to 0 in Frobenius series?

In the Frobenius series method, the constant term c0 in the power series solution cannot be equal to 0. This is because if c0 is 0, then the resulting series becomes an infinite polynomial, which is not a valid solution for the differential equation.

## 2. Can c0 be equal to 0 in any case for Frobenius series?

No, c0 cannot be equal to 0 in any case for the Frobenius series method. This condition is necessary for the convergence of the series and for obtaining a valid solution for the differential equation.

## 3. What is the significance of c0 in Frobenius series?

The constant term c0 in the Frobenius series represents the initial condition of the differential equation. It is the value of the dependent variable at the initial point, and is used to determine the coefficients of the power series solution.

## 4. How is c0 determined in the Frobenius series method?

The value of c0 is determined by substituting the initial conditions of the differential equation into the power series solution. This allows for the coefficients of the series to be calculated and for the solution to be obtained.

## 5. Can c0 be a complex number in Frobenius series?

Yes, c0 can be a complex number in Frobenius series. This is because the power series solution for a complex differential equation can also have complex coefficients, and c0 is just one of these coefficients.