$$ { } ^ { m } ( z _ { 0 } + h ) - f ^ { m } ( z _ { 0 } ) = \frac { m ! } { 2 \pi i } \int _ { c } ^ { 1 } \frac { f ( z ) d z } { ( z - z _ { 0 } - h ) } - \frac { m ! } { 2 \pi i } \int _ { c } \frac { f ( z ) d z } { ( z - z _ { 0 } - h ) ^ { m + 1 } } $$

Question

UpStudy AI · Accepted Answer

The equation you've provided appears to relate to complex analysis, specifically involving the use of contour integrals to express differences in derivatives of a function. Let's break down and interpret each part of the equation to understand its significance and potential applications.

### The Given Equation

$$
f^{(m)}(z_0 + h) - f^{(m)}(z_0) = \frac{m!}{2\pi i} \int_{C_1} \frac{f(z) \, dz}{(z - z_0 - h)} - \frac{m!}{2\pi i} \int_{C} \frac{f(z) \, dz}{(z - z_0 - h)^{m + 1}}
$$

**Assumptions and Clarifications:**

1. **Notation Interpretation:**
   - $$ f^{(m)}(z) $$ denotes the $$ m $$-th derivative of the function $$ f $$ at the point $$ z $$.
   - $$ z_0 $$ is a point in the complex plane around which the function is being analyzed.
   - $$ h $$ is a small increment in the complex plane.
   - $$ C $$ and $$ C_1 $$ are contours (closed paths) in the complex plane, potentially chosen to encircle certain points of interest.

2. **Contextual Foundation:**
   - **Cauchy's Differentiation Formula:** In complex analysis, the $$ m $$-th derivative of an analytic function $$ f $$ at a point $$ z_0 $$ can be expressed as:

$$
     f^{(m)}(z_0) = \frac{m!}{2\pi i} \oint_{C} \frac{f(z)}{(z - z_0)^{m+1}}} \, dz
     $$

where $$ C $$ is a positively oriented, simple closed contour surrounding $$ z_0 $$.

### Interpreting the Equation

1. **Left-Hand Side (LHS):**
   
   $$
   f^{(m)}(z_0 + h) - f^{(m)}(z_0)
   $$

This represents the difference between the $$ m $$-th derivatives of $$ f $$ evaluated at two nearby points $$ z_0 + h $$ and $$ z_0 $$.

2. **Right-Hand Side (RHS):**

The RHS consists of two contour integrals:

$$
   \frac{m!}{2\pi i} \int_{C_1} \frac{f(z) \, dz}{(z - z_0 - h)} - \frac{m!}{2\pi i} \int_{C} \frac{f(z) \, dz}{(z - z_0 - h)^{m + 1}}
   $$

- **First Integral:**
     
     $$
     \frac{m!}{2\pi i} \int_{C_1} \frac{f(z)}{(z - z_0 - h)} \, dz
     $$
     
     This resembles the expression for the first derivative of $$ f $$ at $$ z_0 + h $$ if we consider Cauchy's formula for $$ m = 0 $$, but adjusted by $$ h $$.

- **Second Integral:**
     
     $$
     \frac{m!}{2\pi i} \int_{C} \frac{f(z)}{(z - z_0 - h)^{m + 1}}} \, dz
     $$
     
     This closely aligns with the $$ m $$-th derivative expression from Cauchy's Differentiation Formula, but again shifted by $$ h $$.

3. **Potential Interpretation:**

The equation seems to express how the $$ m $$-th derivative of $$ f $$ changes as the point of evaluation shifts by $$ h $$. It decomposes this change into two integral components:

- The first integral could represent the **incremental change** in the derivative due to the shift $$ h $$.
   - The second integral maintains the structure of the $$ m $$-th derivative but evaluated at the shifted point $$ z_0 + h $$.

4. **Possible Applications:**

- **Analytic Continuation:** Understanding how derivatives behave under shifts in the complex plane.
   - **Perturbation Theory:** Studying small changes in the system or function and their impact on derivatives.
   - **Numerical Analysis:** Providing integral representations that could be useful for numerical approximations of derivative differences.

### Verifying the Equation

To ensure the equation holds, one could:

1. **Start from Cauchy's Differentiation Formula:** Express both $$ f^{(m)}(z_0 + h) $$ and $$ f^{(m)}(z_0) $$ using the integral representations and subtract them.

2. **Analyze the Contours $$ C $$ and $$ C_1 $$:** Ensure that these contours are defined appropriately (e.g., both encircling the necessary poles) to validate the integral expressions.

3. **Simplify the Difference:** Through algebraic manipulation, check if the difference of the two integral representations indeed equates to the RHS of your original equation.

### Conclusion

The equation you've presented is a sophisticated expression likely rooted in complex analysis, relating to how higher-order derivatives of an analytic function change with a shift in the point of evaluation. To provide a more precise explanation or application, additional context about the function $$ f $$, the contours $$ C $$ and $$ C_1 $$, and the parameters involved would be helpful.

If you have a specific question about this equation—such as its derivation, application, or properties—please provide more details, and I'd be glad to assist further!
The equation relates the difference between the $$ m $$-th derivatives of a function at two nearby points to contour integrals of the function. It shows how the derivative changes with a small shift $$ h $$ in the complex plane.

\( { } ^ { m } ( z _ { 0 } + h ) - f ^ { m } ( z _ { 0 } ) = \frac { m ! } { 2 \pi i } \int _ { c } ^ { 1 } \frac { f ( z ) d z } { ( z - z _ { 0 } - h ) } - \frac { m ! } { 2 \pi i } \int _ { c } \frac { f ( z ) d z } { ( z - z _ { 0 } - h ) ^ { m + 1 } } \)

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions