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

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UpStudy AI · Accepted Answer

To analyze the given equation, we will break it down step by step. The equation is:

$$
f^{m}(z_{0} + h) - f^{m}(z_{0}) = \frac{m!}{2 \pi i} \int_{c} \frac{f(z) \, dz}{(z - z_{0} - h)} - \frac{m!}{2 \pi i} \int_{c} \frac{f(z) \, dz}{(z - z_{0} - h)^{m + 1}}
$$

### Step 1: Understanding the Terms

1. **Left Side**: 
   - $$ f^{m}(z_{0} + h) $$ and $$ f^{m}(z_{0}) $$ represent the $$ m $$-th derivative of the function $$ f $$ evaluated at $$ z_{0} + h $$ and $$ z_{0} $$, respectively.

2. **Right Side**: 
   - The right side consists of two integrals, both involving the function $$ f(z) $$ and a contour $$ c $$ around the point $$ z_{0} $$.

### Step 2: Analyzing the Integrals

- The first integral:
  $$
  \frac{m!}{2 \pi i} \int_{c} \frac{f(z) \, dz}{(z - z_{0} - h)}
  $$
  represents a residue calculation, which is related to the value of $$ f(z) $$ at the point $$ z_{0} + h $$.

- The second integral:
  $$
  -\frac{m!}{2 \pi i} \int_{c} \frac{f(z) \, dz}{(z - z_{0} - h)^{m + 1}}
  $$
  accounts for higher-order terms in the Taylor expansion of $$ f $$ around $$ z_{0} + h $$.

### Step 3: Interpretation

The equation essentially states that the difference in the $$ m $$-th derivatives of $$ f $$ at two points $$ z_{0} + h $$ and $$ z_{0} $$ can be expressed in terms of contour integrals involving $$ f(z) $$. This is a result of the Cauchy integral formula, which relates the values of analytic functions and their derivatives to integrals over contours.

### Conclusion

This equation is a representation of how the $$ m $$-th derivative of a function can be approximated using contour integrals. It highlights the relationship between the local behavior of the function around a point and its global properties as captured by the integrals.

If you need further calculations or specific evaluations, please provide additional details or context!
The equation relates the difference in the $$ m $$-th derivatives of a function $$ f $$ at two points to contour integrals of $$ f(z) $$.

Upstudy AI Solution

Answer

Solution

Step 1: Understanding the Terms

Step 2: Analyzing the Integrals

Step 3: Interpretation

Conclusion

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